Sequences and Series

Definition of Sequence and Series

Ordered Lists of Numbers and Their Sums

In mathematics, we frequently encounter collections of numbers that are arranged in a particular order, often following a predictable rule or pattern. These ordered lists are fundamental objects known as sequences. When we take the numbers in a sequence and add them together, we create a new object called a series. The study of sequences and series is a core topic in algebra and analysis, providing tools to understand patterns and sums.

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Definition of a Sequence

A sequence is formally defined as an ordered list of numbers. The individual numbers that constitute the sequence are called the terms of the sequence. The order in which the terms appear is crucial; changing the order results in a different sequence. Terms in a sequence are typically separated by commas.

Sequences can be classified based on the number of terms they contain:

• Finite Sequence: A finite sequence is a sequence that has a specific, limited number of terms. The list of numbers comes to an end.

  Example: $2, 4, 6, 8$ - This is a finite sequence consisting of the first four positive even numbers.

  Example: $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ - This is a finite sequence of the reciprocals of the first four positive integers.

  Example: The list of marks obtained by students in a class in a test.

• Infinite Sequence: An infinite sequence is a sequence that continues indefinitely, having an unlimited number of terms. This is usually indicated by appending three dots (... ) at the end of the list.

  Example: $1, 3, 5, 7, \ldots$ - This represents the infinite sequence of all positive odd numbers.

  Example: $1, 2, 4, 8, 16, \ldots$ - This is the infinite sequence where each term is a power of 2.

  Example: $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots$ - This is an infinite sequence where each term is a power of $\frac{1}{2}$.

The terms of a sequence are often denoted using subscript notation to indicate their position in the sequence. For example, the first term is denoted by $a_1$, the second term by $a_2$, the third term by $a_3$, and generally, the $n$-th term (or the term at position $n$) is denoted by $a_n$. The sequence itself can be represented as $\{a_n\}$ or $(a_n)_{n=1}^{\infty}$.

Many sequences follow a specific rule or formula that defines the $n$-th term $a_n$ as a function of $n$. This formula allows us to find any term in the sequence simply by knowing its position $n$. Such a formula is called the general term or the n-th term formula of the sequence.

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Definition of a Series

A series is defined as the sum of the terms of a sequence. If we have a sequence $a_1, a_2, a_3, \ldots$, the corresponding series is formed by adding these terms together: $a_1 + a_2 + a_3 + \ldots$.

Just like sequences, series can be finite or infinite, depending on whether the sequence they are derived from is finite or infinite.

• Finite Series: A finite series is the sum of the terms of a finite sequence. It has a definite number of terms, and its sum can be calculated.

  Example: $2 + 4 + 6 + 8$ - This is a finite series which is the sum of the first four positive even numbers. The sum is $2+4+6+8=20$.

  Example: $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}$ - This is a finite series. The sum is $1 + 0.5 + 0.333\ldots + 0.25 = 2.08333\ldots$

• Infinite Series: An infinite series is the sum of the terms of an infinite sequence. It has an infinite number of terms. Calculating the "sum" of an infinite series is a more complex concept (convergence) and depends on whether the sum approaches a finite value as more and more terms are added.

  Example: $1 + 3 + 5 + 7 + \ldots$ - This is an infinite series. As we add more terms, the sum keeps increasing without bound. Such a series is said to diverge.

  Example: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots$ - This is an infinite series. As we add more terms, the sum gets closer and closer to 2. This series is said to converge to 2.

The sum of the first $n$ terms of a sequence $a_1, a_2, a_3, \ldots$ is called the partial sum and is typically denoted by $S_n$.

Using summation notation (Sigma notation), the sum of the first $n$ terms can be written concisely as:

$$ S_n = \sum_{i=1}^{n} a_i $$

Here, $\sum$ (the Greek capital letter sigma) means "summation", $a_i$ is the formula for the terms, and $i=1$ to $n$ indicates that we are summing the terms starting from the first term ($i=1$) up to the $n$-th term ($i=n$).

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The study of sequences and series involves identifying the pattern (e.g., arithmetic, geometric, etc.), finding a formula for the general term ($a_n$), deriving formulas for the sum of the first $n$ terms ($S_n$), and for infinite series, determining if the sum converges to a finite value.

Arithmetic Progression (AP): Definition, General Term, Properties

Sequences with a Constant Difference

In the study of sequences, certain types of patterns occur very frequently and are therefore given special names. One such important type is the Arithmetic Progression (AP). An AP is distinguished by a very simple characteristic: the difference between any term and the term immediately before it is always the same throughout the sequence. This constant difference is known as the common difference.

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Definition of Arithmetic Progression

A sequence of numbers, $a_1, a_2, a_3, \ldots, a_n, \ldots$, is defined as an Arithmetic Progression (AP) if the difference between consecutive terms is constant. Mathematically, this means that for every positive integer $n$, the difference between the $(n+1)$-th term and the $n$-th term is the same fixed value. This fixed value is denoted by $d$ and is called the common difference of the AP.

Thus, a sequence is an AP if $a_{n+1} - a_n = d$ for all $n \ge 1$.

If the first term of an AP is denoted by $a$ (instead of $a_1$), and the common difference is $d$, then the terms of the AP can be expressed in terms of $a$ and $d$ as follows:

• First term ($n=1$): $a_1 = a$
• Second term ($n=2$): $a_2 = a_1 + d = a + d$
• Third term ($n=3$): $a_3 = a_2 + d = (a+d) + d = a + 2d$
• Fourth term ($n=4$): $a_4 = a_3 + d = (a+2d) + d = a + 3d$
• And so on...

So, the terms of an AP starting with $a$ and having a common difference $d$ are:

Examples of Arithmetic Progressions:

• $2, 5, 8, 11, 14, \ldots$: Here, $a=2$ and the common difference $d = 5 - 2 = 3$. We can check this for other consecutive terms: $8-5=3$, $11-8=3$, etc. This is an AP with $d=3$.
• $10, 8, 6, 4, 2, \ldots$: Here, $a=10$ and the common difference $d = 8 - 10 = -2$. Checking: $6-8=-2$, $4-6=-2$, etc. This is an AP with $d=-2$.
• $1, 1.5, 2, 2.5, 3, \ldots$: Here, $a=1$ and the common difference $d = 1.5 - 1 = 0.5$. Checking: $2-1.5=0.5$, $2.5-2=0.5$, etc. This is an AP with $d=0.5$.
• $\frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, \ldots$: Here, $a=\frac{1}{2}$ and the common difference $d = 1 - \frac{1}{2} = \frac{1}{2}$. Checking: $\frac{3}{2} - 1 = \frac{1}{2}$, $2 - \frac{3}{2} = \frac{1}{2}$, etc. This is an AP with $d=\frac{1}{2}$.
• $7, 7, 7, 7, \ldots$: Here, $a=7$ and the common difference $d = 7 - 7 = 0$. This is an AP with $d=0$.

The common difference $d$ can be positive, negative, or zero.

• If $d > 0$, the AP is increasing.
• If $d < 0$, the AP is decreasing.
• If $d = 0$, the AP is constant (all terms are the same).

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General Term (n-th Term) of an AP

The general term, also known as the $n$-th term, of an Arithmetic Progression is a formula that allows us to determine the value of any term in the sequence if we know its position $n$, the first term $a$, and the common difference $d$. Looking at the pattern of the terms derived from the definition:

• $a_1 = a = a + (1-1)d$
• $a_2 = a + d = a + (2-1)d$
• $a_3 = a + 2d = a + (3-1)d$
• $a_4 = a + 3d = a + (4-1)d$
• ...

We can observe a clear pattern: the coefficient of $d$ is always one less than the term number $n$. Thus, the formula for the $n$-th term of an AP is:

where:

• $a_n$ is the value of the $n$-th term.
• $a$ is the first term ($a_1$).
• $d$ is the common difference.
• $n$ is the position of the term in the sequence (a positive integer, $n \ge 1$).

Derivation:

We start with the recursive definition of an AP: $a_{k+1} = a_k + d$ for $k \ge 1$, and $a_1 = a$.

$a_2 = a_1 + d$

$a_3 = a_2 + d = (a_1 + d) + d = a_1 + 2d$

$a_4 = a_3 + d = (a_1 + 2d) + d = a_1 + 3d$

By observing this pattern, we can see that to get to the $n$-th term ($a_n$) starting from the first term ($a_1$), we need to add the common difference $d$ a total of $(n-1)$ times. Therefore, the $n$-th term is $a_n = a_1 + (n-1)d$. If we use $a$ to denote the first term $a_1$, the formula is $a_n = a + (n-1)d$.

Example

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Properties of an AP

Arithmetic Progressions have several interesting and useful properties:

1. Adding or Subtracting a Constant: If each term of an AP is increased or decreased by the same constant number $k$, the resulting sequence is also an AP, and its common difference remains the same as the original AP.

   Example: AP: $2, 5, 8, 11$ with $d=3$. Add 10 to each term: $2+10, 5+10, 8+10, 11+10$, which is $12, 15, 18, 21$. This new sequence is also an AP with a common difference of $15-12=3$, which is the same as the original $d$.

2. Multiplying or Dividing by a Non-Zero Constant: If each term of an AP is multiplied or divided by the same non-zero constant number $k$, the resulting sequence is also an AP. The common difference of the new AP will be the original common difference multiplied or divided by $k$.

   Example: AP: $2, 5, 8, 11$ with $d=3$. Multiply each term by 2: $2 \times 2, 5 \times 2, 8 \times 2, 11 \times 2$, which is $4, 10, 16, 22$. This new sequence is an AP with common difference $10-4=6$. The new difference is $3 \times 2 = 6$.

   Example: AP: $10, 8, 6, 4$ with $d=-2$. Divide each term by 2: $\frac{10}{2}, \frac{8}{2}, \frac{6}{2}, \frac{4}{2}$, which is $5, 4, 3, 2$. This new sequence is an AP with common difference $4-5=-1$. The new difference is $\frac{-2}{2} = -1$.

3. Common Difference Definition: The fundamental property: $a_{n+1} - a_n = d$ for all $n \ge 1$. This property is used to identify if a given sequence is an AP.
4. Arithmetic Mean Property: For any three consecutive terms $a_{n-1}, a_n, a_{n+1}$ in an AP (where $n \ge 2$), the middle term $a_n$ is the arithmetic mean of the other two terms. That is, $a_n = \frac{a_{n-1} + a_{n+1}}{2}$. This comes directly from the definition: $a_n - a_{n-1} = d$ and $a_{n+1} - a_n = d$. So, $a_n - a_{n-1} = a_{n+1} - a_n$. Rearranging, $2a_n = a_{n-1} + a_{n+1}$, which gives $a_n = \frac{a_{n-1} + a_{n+1}}{2}$. If $a, b, c$ are in AP, then $b = \frac{a+c}{2}$.
5. Sum of Equidistant Terms: In a finite AP with $n$ terms, the sum of any two terms that are equidistant from the beginning and the end is constant and equal to the sum of the first and the last terms. If the AP is $a_1, a_2, \ldots, a_n$, then $a_1 + a_n = a_2 + a_{n-1} = a_3 + a_{n-2} = \ldots = a_k + a_{n-k+1}$.

   Example: AP: $3, 7, 11, 15, 19, 23$. $a_1=3, a_6=23$. Sum = $3+23=26$. $a_2=7, a_5=19$. Sum = $7+19=26$. $a_3=11, a_4=15$. Sum = $11+15=26$.

These properties are useful for solving problems related to arithmetic progressions.

Sum of First n Terms of an AP

Finding the Sum of an Arithmetic Series

When we consider a finite Arithmetic Progression (AP), which is an ordered list of numbers with a constant difference between consecutive terms, the sum of these terms forms an arithmetic series. A common problem related to APs is calculating the total sum when a specific number of initial terms are added together. We denote the sum of the first $n$ terms of an AP by $S_n$.

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Formula for the Sum of First n Terms

Let's consider an Arithmetic Progression with the first term $a$ and a common difference $d$. The first $n$ terms of this AP are $a, a+d, a+2d, \ldots$. The $n$-th term is given by $a_n = a + (n-1)d$. If we denote the $n$-th term (which is the last term in this finite series) by $l$, then $l = a + (n-1)d$.

The sum of the first $n$ terms, $S_n$, is the sum of these terms:

$$ S_n = a + (a+d) + (a+2d) + \ldots + (a+(n-2)d) + (a+(n-1)d) \quad \text{... (1)} $$

A clever way to find a formula for $S_n$ is to write the same sum in reverse order. If the last term is $l$, the term before it is $l-d$, the term before that is $l-2d$, and so on, until the first term $a$, which can be written as $l-(n-1)d$. So, the sum in reverse order is:

$$ S_n = l + (l-d) + (l-2d) + \ldots + (l-(n-2)d) + (l-(n-1)d) \quad \text{... (2)} $$

Now, let's add Equation (1) and Equation (2) term by term, vertically lining up the corresponding terms:

$$ S_n \quad = \quad a \quad + \quad (a+d) \quad + \quad (a+2d) \quad + \quad \ldots \quad + \quad (a+(n-1)d) $$ $$ S_n \quad = \quad l \quad + \quad (l-d) \quad + \quad (l-2d) \quad + \quad \ldots \quad + \quad (l-(n-1)d) $$ $$ \rule{5cm}{0.4pt} $$ $$ S_n + S_n = (a+l) + ((a+d) + (l-d)) + ((a+2d) + (l-2d)) + \ldots + ((a+(n-1)d) + (l-(n-1)d)) $$ $$ 2S_n = (a+l) + (a+l) + (a+l) + \ldots + (a+l) $$

In the original series, there are $n$ terms. When we added the two series, we formed $n$ pairs, and the sum of each pair is $(a+l)$. Therefore, the sum $2S_n$ is equal to $n$ times the quantity $(a+l)$.

$$ 2S_n = n \times (a+l) $$

Dividing both sides by 2, we get the formula for $S_n$ in terms of the first term and the last term:

where:

• $S_n$ is the sum of the first $n$ terms.
• $n$ is the number of terms.
• $a$ is the first term ($a_1$).
• $l$ is the last term ($a_n$).

Often, the last term $l$ is not given directly, but the common difference $d$ is known. In this case, we can use the formula for the $n$-th term, $l = a_n = a + (n-1)d$, and substitute it into the formula $S_n = \frac{n}{2}(a + l)$:

$$ S_n = \frac{n}{2}(a + [a + (n-1)d]) $$ $$ S_n = \frac{n}{2}[a + a + (n-1)d] $$

Simplifying the expression inside the square brackets:

where:

• $S_n$ is the sum of the first $n$ terms.
• $n$ is the number of terms.
• $a$ is the first term ($a_1$).
• $d$ is the common difference.

Both formulas are equivalent and can be used depending on the information provided. The first formula is useful when the first and last terms are known, while the second formula is useful when the first term, common difference, and number of terms are known.

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Examples

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Sum of First n Positive Integers

A particularly important and common Arithmetic Progression is the sequence of the first $n$ positive integers: $1, 2, 3, \ldots, n$.

For this specific AP:

• The first term is $a = 1$.
• The common difference is $d = 2 - 1 = 1$.
• The number of terms is $n$.
• The last term is $l = n$.

We can find the sum of these terms, $S_n$, using either sum formula.

Using the formula $S_n = \frac{n}{2}(a + l)$:

$$ S_n = \frac{n}{2}(1 + n) $$ $$ S_n = \frac{n(n + 1)}{2} $$

Using the formula $S_n = \frac{n}{2}[2a + (n-1)d]$:

$$ S_n = \frac{n}{2}[2(1) + (n-1)(1)] $$ $$ S_n = \frac{n}{2}[2 + n - 1] $$ $$ S_n = \frac{n}{2}(n + 1) $$ $$ S_n = \frac{n(n + 1)}{2} $$

Both formulas give the same simple and famous result:

Example: Find the sum of the first 100 positive integers.

Here, $n=100$. Using the formula:

$$ S_{100} = \frac{100(100 + 1)}{2} = \frac{100 \times 101}{2} = \frac{10100}{2} $$ $$ S_{100} = 5050 $$

The sum of the first 100 positive integers is 5050.

Understanding the formula for the sum of an AP is crucial for solving problems involving the total value of terms in a sequence that follows an arithmetic pattern. It also has applications in various fields like finance, physics, and computer science.

Arithmetic Mean (AM)

The Average of Numbers

The term Arithmetic Mean (AM) refers to the most commonly used type of average. It provides a measure of central tendency for a set of numbers. While the concept of AM applies to any set of numbers, it holds a special significance and relationship with Arithmetic Progressions (APs).

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Definition of Arithmetic Mean

For a given set of $n$ numbers, say $x_1, x_2, \ldots, x_n$, their Arithmetic Mean (AM) is calculated by finding the sum of all the numbers and then dividing this sum by the total count of the numbers ($n$).

The formula for the Arithmetic Mean of $n$ numbers is:

Using summation notation, this can be written concisely as:

$$ \text{AM} = \frac{\sum_{i=1}^{n} x_i}{n} $$

where $\sum_{i=1}^{n} x_i$ represents the sum of the numbers from $x_1$ to $x_n$.

Example: Find the Arithmetic Mean of the numbers $10, 20,$ and $30$.

Here, we have $n=3$ numbers. The sum is $10 + 20 + 30 = 60$.

The AM is $\frac{\text{Sum of numbers}}{\text{Count of numbers}} = \frac{60}{3} = 20$.

The Arithmetic Mean of 10, 20, and 30 is 20.

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Arithmetic Mean in the Context of AP

One of the key properties of an Arithmetic Progression relates to the arithmetic mean of its terms. If you consider any three consecutive terms of an AP, say $a, b,$ and $c$, there is a specific relationship between the middle term $b$ and the other two terms $a$ and $c$.

By the definition of an Arithmetic Progression, the difference between consecutive terms is constant (the common difference, $d$). So, if $a, b,$ and $c$ are consecutive terms in an AP, then:

Since both differences are equal to $d$, they must be equal to each other:

$$ b - a = c - b $$

Now, rearrange this equation to solve for $b$. Add $b$ to both sides:

$$ b - a + b = c - b + b $$ $$ 2b - a = c $$

Add $a$ to both sides:

$$ 2b = a + c $$

Divide both sides by 2:

$$ b = \frac{a+c}{2} $$

This result shows that the middle term ($b$) of any three consecutive terms in an AP is the Arithmetic Mean of the first ($a$) and the third ($c$) terms.

Example: Consider the AP $5, 8, 11$. Here $a=5, b=8, c=11$. The middle term is $8$. Let's calculate the AM of the other two terms, 5 and 11:

$$ \text{AM of 5 and 11} = \frac{5 + 11}{2} = \frac{16}{2} = 8 $$

The AM of 5 and 11 is 8, which is indeed the middle term of the AP.

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Inserting Arithmetic Means Between Two Numbers

A common problem is to insert a specific number of terms between two given numbers such that the entire sequence forms an Arithmetic Progression. The numbers inserted are called arithmetic means between the two given numbers.

Suppose we want to insert $n$ arithmetic means between two numbers $A$ and $B$. Let these $n$ means be $m_1, m_2, \ldots, m_n$. The sequence will look like this:

$$ A, m_1, m_2, \ldots, m_n, B $$

For this sequence to be an AP, there must be a constant common difference, say $d$. In this sequence, $A$ is the first term and $B$ is the last term. How many terms are there in total? We have the initial term $A$, the final term $B$, and $n$ terms inserted in between. So, the total number of terms in the AP is $n+2$.

Let the first term of this AP be $a_{1} = A$. The last term is $a_{n+2} = B$. The total number of terms is $N = n+2$.

Using the formula for the $N$-th term of an AP, $a_N = a_1 + (N-1)d$:

$$ a_{n+2} = A + ((n+2) - 1)d $$ $$ B = A + (n+1)d $$

Now, we can solve this equation for the common difference $d$:

$$ B - A = (n+1)d $$ $$ d = \frac{B - A}{n + 1} $$

Once the common difference $d$ is calculated, we can find the inserted arithmetic means using the definition of an AP:

$$ m_1 = A + d $$ $$ m_2 = A + 2d $$ $$ m_3 = A + 3d $$ $$ \ldots $$ $$ m_n = A + nd $$

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The Arithmetic Mean is a fundamental concept in statistics and sequence theory, particularly useful for understanding the structure of Arithmetic Progressions and for interpolating terms within an AP.

Geometric Progression (GP): Definition, General Term, Properties

Sequences with a Constant Ratio

Similar to Arithmetic Progressions, Geometric Progressions (GPs) are another fundamental type of sequence characterized by a constant relationship between consecutive terms. While an AP has a constant *difference*, a GP has a constant *ratio*. This constant ratio is called the common ratio.

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Definition of Geometric Progression

A sequence of numbers, $a_1, a_2, a_3, \ldots, a_n, \ldots$, with non-zero terms is defined as a Geometric Progression (GP) if the ratio of any term to its immediately preceding term is a constant value. This constant value is denoted by $r$ and is called the common ratio of the GP.

Thus, a sequence is a GP if $\frac{a_{n+1}}{a_n} = r$ for all $n \ge 1$, where $r$ is a non-zero constant. This can also be written recursively as $a_{n+1} = a_n \cdot r$.

If the first term of a GP is denoted by $a$ (instead of $a_1$) and the common ratio is $r$, then the terms of the GP can be expressed in terms of $a$ and $r$ by repeatedly multiplying by the common ratio:

• First term ($n=1$): $a_1 = a$
• Second term ($n=2$): $a_2 = a_1 \cdot r = a \cdot r$
• Third term ($n=3$): $a_3 = a_2 \cdot r = (ar) \cdot r = ar^2$
• Fourth term ($n=4$): $a_4 = a_3 \cdot r = (ar^2) \cdot r = ar^3$
• And so on...

So, the terms of a GP starting with $a$ and having a common ratio $r$ are:

Examples of Geometric Progressions:

• $2, 6, 18, 54, \ldots$: Here, $a=2$ and the common ratio $r = \frac{a_2}{a_1} = \frac{6}{2} = 3$. Check with other terms: $\frac{18}{6} = 3$, $\frac{54}{18} = 3$. This is a GP with $r=3$.
• $10, 5, 2.5, 1.25, \ldots$: Here, $a=10$ and the common ratio $r = \frac{a_2}{a_1} = \frac{5}{10} = \frac{1}{2}$. Check: $\frac{2.5}{5} = 0.5 = \frac{1}{2}$, $\frac{1.25}{2.5} = 0.5 = \frac{1}{2}$. This is a GP with $r=\frac{1}{2}$.
• $1, -1, 1, -1, \ldots$: Here, $a=1$ and the common ratio $r = \frac{a_2}{a_1} = \frac{-1}{1} = -1$. Check: $\frac{1}{-1}=-1$, $\frac{-1}{1}=-1$. This is a GP with $r=-1$. The terms alternate in sign.
• $3, 3, 3, 3, \ldots$: Here, $a=3$ and the common ratio $r = \frac{a_2}{a_1} = \frac{3}{3} = 1$. This is a GP with $r=1$. All terms are the same.
• $\frac{1}{4}, -\frac{1}{2}, 1, -2, 4, \ldots$: Here, $a=\frac{1}{4}$ and the common ratio $r = \frac{-1/2}{1/4} = -\frac{1}{2} \times \frac{4}{1} = -2$. Check: $\frac{1}{-1/2} = -2$, $\frac{-2}{1}=-2$. This is a GP with $r=-2$.

The common ratio $r$ can be positive or negative, greater than 1, less than -1, or between -1 and 1 (but not zero).

• If $|r| > 1$, the terms of the GP increase in magnitude (unless $r=-1$).
• If $|r| < 1$, the terms of the GP decrease in magnitude, approaching 0.
• If $r = 1$, the GP is constant.
• If $r = -1$, the GP alternates between $a$ and $-a$.

Note: By definition, the common ratio $r$ cannot be zero. If $a_1 = 0$, and $r$ is non-zero, the sequence is $0, 0, 0, \ldots$ which is technically an AP with $d=0$ and a GP with $r$ being any non-zero number (but the ratio $0/0$ is undefined). Usually, we consider GPs where the first term $a \neq 0$ and $r \neq 0$.

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General Term (n-th Term) of a GP

The general term, or $n$-th term, of a Geometric Progression, denoted by $a_n$, is a formula that allows us to find the value of any term in the sequence using its position $n$, the first term $a$, and the common ratio $r$. From the pattern we observed in the definition:

• $a_1 = a = ar^0 = ar^{1-1}$
• $a_2 = ar = ar^1 = ar^{2-1}$
• $a_3 = ar^2 = ar^{3-1}$
• $a_4 = ar^3 = ar^{4-1}$
• ...

The exponent of $r$ is always one less than the term number $n$. Thus, the formula for the $n$-th term of a GP is:

where:

• $a_n$ is the value of the $n$-th term.
• $a$ is the first term ($a_1$).
• $r$ is the common ratio.
• $n$ is the position of the term in the sequence (a positive integer, $n \ge 1$).

Derivation:

We start with the recursive definition of a GP: $a_{k+1} = a_k \cdot r$ for $k \ge 1$, and $a_1 = a$.

$a_2 = a_1 \cdot r$

$a_3 = a_2 \cdot r = (a_1 \cdot r) \cdot r = a_1 \cdot r^2$

$a_4 = a_3 \cdot r = (a_1 \cdot r^2) \cdot r = a_1 \cdot r^3$

By observing this pattern, we see that to get to the $n$-th term ($a_n$) starting from the first term ($a_1$), we multiply by the common ratio $r$ a total of $(n-1)$ times. Therefore, the $n$-th term is $a_n = a_1 \cdot r^{n-1}$. If we use $a$ to denote the first term $a_1$, the formula is $a_n = a r^{n-1}$.

Example

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Properties of a GP

Geometric Progressions possess several distinctive properties:

1. Multiplying or Dividing by a Non-Zero Constant: If each term of a GP is multiplied or divided by the same non-zero constant $k$, the resulting sequence is also a GP. The common ratio of the new GP will be the same as the common ratio of the original GP.

   Example: GP: $2, 6, 18$ with $r=3$. Multiply each term by 5: $2 \times 5, 6 \times 5, 18 \times 5$, which is $10, 30, 90$. This new sequence is a GP with common ratio $\frac{30}{10}=3$, which is the same as the original $r$. Divide each term by 2: $\frac{2}{2}, \frac{6}{2}, \frac{18}{2}$, which is $1, 3, 9$. This new sequence is a GP with common ratio $\frac{3}{1}=3$, same as the original $r$.

2. Common Ratio Definition: The fundamental property: $\frac{a_{n+1}}{a_n} = r$ for all $n \ge 1$. This property is used to verify if a given sequence is a GP.
3. Geometric Mean Property: For any three consecutive terms $a_{n-1}, a_n, a_{n+1}$ in a GP (where $n \ge 2$), if none of the terms are zero, the square of the middle term $a_n$ is equal to the product of the other two terms $a_{n-1}$ and $a_{n+1}$. That is, $a_n^2 = a_{n-1} \cdot a_{n+1}$. This comes directly from the definition: $\frac{a_n}{a_{n-1}} = r$ and $\frac{a_{n+1}}{a_n} = r$. So, $\frac{a_n}{a_{n-1}} = \frac{a_{n+1}}{a_n}$. Cross-multiplying gives $a_n \cdot a_n = a_{n-1} \cdot a_{n+1}$, or $a_n^2 = a_{n-1} a_{n+1}$. If $a, b, c$ are in GP (and non-zero), then $b^2 = ac$. The middle term $b$ is the geometric mean of $a$ and $c$ (specifically, $b = \sqrt{ac}$ if all terms are positive, or $b = \pm \sqrt{ac}$).
4. Product of Equidistant Terms: In a finite GP with $n$ terms, the product of any two terms that are equidistant from the beginning and the end is constant and equal to the product of the first and the last terms. If the GP is $a_1, a_2, \ldots, a_n$, then $a_1 \times a_n = a_2 \times a_{n-1} = a_3 \times a_{n-2} = \ldots = a_k \times a_{n-k+1}$.

   Example: GP: $2, 6, 18, 54, 162, 486$. $a_1=2, a_6=486$. Product = $2 \times 486 = 972$. $a_2=6, a_5=162$. Product = $6 \times 162 = 972$. $a_3=18, a_4=54$. Product = $18 \times 54 = 972$.

These properties are useful for identifying GPs, finding missing terms, and solving problems involving sequences with a geometric pattern.

Geometric Mean (GM)

The Geometric Average

In addition to the arithmetic mean, another important type of average is the Geometric Mean (GM). While the AM is suitable for averaging quantities that are additive in nature (like terms in an AP), the GM is particularly useful for averaging quantities that are related by multiplication or represent rates of change, such as average growth rates over time or terms in a Geometric Progression.

The definition of the geometric mean requires the numbers to be positive.

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Definition of Geometric Mean

For a set of $n$ positive numbers, say $x_1, x_2, \ldots, x_n$, their Geometric Mean (GM) is defined as the $n$-th root of their product.

The formula for the Geometric Mean of $n$ positive numbers is:

Using product notation (Pi notation), this can be written concisely as:

$$ \text{GM} = \sqrt[n]{\prod_{i=1}^{n} x_i} $$

where $\prod_{i=1}^{n} x_i$ represents the product of the numbers from $x_1$ to $x_n$.

Important Note: The geometric mean is only defined for a set of positive numbers. If any number in the set is zero or negative, the geometric mean is not typically calculated or defined in this standard way.

Examples:

• Find the Geometric Mean of $2$ and $8$. Here, $n=2$ positive numbers. $$ \text{GM} = \sqrt[2]{2 \times 8} = \sqrt{16} = 4 $$ The GM of 2 and 8 is 4.
• Find the Geometric Mean of $1, 3,$ and $9$. Here, $n=3$ positive numbers. $$ \text{GM} = \sqrt[3]{1 \times 3 \times 9} = \sqrt[3]{27} = 3 $$ The GM of 1, 3, and 9 is 3.

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Geometric Mean in the Context of GP

Just as the arithmetic mean is related to consecutive terms in an AP, the geometric mean has a special connection to consecutive terms in a GP. If you consider any three consecutive positive terms of a GP, say $a, b,$ and $c$, the middle term $b$ is the geometric mean of the other two terms $a$ and $c$.

If $a, b,$ and $c$ are consecutive positive terms in a Geometric Progression, then by the definition of a GP, the ratio of consecutive terms is constant (the common ratio, $r$). So:

Since both ratios are equal to $r$, they must be equal to each other:

$$ \frac{b}{a} = \frac{c}{b} $$

Now, cross-multiply this equation:

$$ b \times b = a \times c $$ $$ b^2 = ac $$

Since we assumed $a, b,$ and $c$ are positive terms, $b^2$ is positive. Taking the positive square root of both sides:

$$ \sqrt{b^2} = \sqrt{ac} $$ $$ b = \sqrt{ac} $$

This result shows that the middle term ($b$) of any three consecutive positive terms in a GP is the Geometric Mean of the first ($a$) and the third ($c$) terms.

Example: Consider the GP $3, 6, 12$. Here $a=3, b=6, c=12$. All terms are positive. The middle term is 6. Let's calculate the GM of the other two terms, 3 and 12:

$$ \text{GM of 3 and 12} = \sqrt{3 \times 12} = \sqrt{36} = 6 $$

The GM of 3 and 12 is 6, which is indeed the middle term of the GP.

Note: If the terms of the GP are not restricted to be positive (e.g., $-2, -4, -8$), the relationship $b^2 = ac$ ($(-4)^2 = (-2)(-8) \implies 16=16$) still holds, but $b$ might not be equal to $\sqrt{ac}$ if we take the positive square root convention for $\sqrt{}$. However, $b = \pm \sqrt{ac}$. For positive terms, $b = \sqrt{ac}$ holds.

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Inserting Geometric Means Between Two Numbers

Similar to inserting arithmetic means, we can insert a specific number of terms between two given positive numbers such that the resulting sequence forms a Geometric Progression. The numbers inserted are called geometric means between the two given numbers.

Suppose we want to insert $n$ geometric means between two positive numbers $A$ and $B$. Let these $n$ means be $m_1, m_2, \ldots, m_n$. The sequence will be:

$$ A, m_1, m_2, \ldots, m_n, B $$

For this sequence to be a GP, there must be a constant common ratio, say $r$. In this sequence, $A$ is the first term and $B$ is the last term. The total number of terms in this GP is the initial term $A$, the final term $B$, plus the $n$ inserted terms, giving a total of $n+2$ terms.

Let the first term of this GP be $a_{1} = A$. The last term is $a_{n+2} = B$. The total number of terms is $N = n+2$.

Using the formula for the $N$-th term of a GP, $a_N = a_1 r^{N-1}$:

$$ a_{n+2} = A r^{(n+2) - 1} $$ $$ B = A r^{n+1} $$

Now, we solve this equation for the common ratio $r$. Divide both sides by $A$ (since $A$ is positive, $A \neq 0$):

$$ \frac{B}{A} = r^{n+1} $$

To find $r$, take the $(n+1)$-th root of both sides:

$$ r = \left(\frac{B}{A}\right)^{\frac{1}{n+1}} $$

Or using radical notation:

$$ r = \sqrt[n+1]{\frac{B}{A}} $$

Since $A$ and $B$ are positive, $\frac{B}{A}$ is positive, and the real $(n+1)$-th root exists and is unique (assuming we are looking for a real common ratio to get real geometric means). If $n+1$ is even and $B/A$ is positive, there are two real roots ($\pm r$), but typically we consider the positive root to get a straightforward sequence of positive geometric means between positive numbers.

Once the common ratio $r$ is calculated, we can find the inserted geometric means using the definition of a GP, starting from the first term $A$:

$$ m_1 = A \cdot r $$ $$ m_2 = m_1 \cdot r = A \cdot r^2 $$ $$ m_3 = m_2 \cdot r = A \cdot r^3 $$ $$ \ldots $$ $$ m_n = A \cdot r^n $$

Example

Answer:

The geometric mean is a distinct type of average used when multiplicative relationships are important. It is essential for understanding Geometric Progressions and other processes involving exponential growth or decay.

Relationship Between AM and GM

Comparing Different Types of Averages

We have discussed two types of averages: the Arithmetic Mean (AM) and the Geometric Mean (GM). The AM is the standard average calculated by summing and dividing, while the GM is calculated as the $n$-th root of the product for $n$ positive numbers. For a set of positive numbers, there exists a fundamental and highly important relationship between their AM and GM. This relationship is expressed as an inequality known as the AM-GM inequality.

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The AM-GM Inequality

The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for any set of $n$ non-negative real numbers $x_1, x_2, \ldots, x_n$, the arithmetic mean is greater than or equal to the geometric mean.

Mathematically, for $x_i \ge 0$ for all $i=1, 2, \ldots, n$:

In terms of the formulas for AM and GM:

The inequality becomes a strict equality (AM = GM) if and only if all the numbers in the set are equal to each other ($x_1 = x_2 = \ldots = x_n$). If the numbers are not all equal, the AM is strictly greater than the GM (AM > GM).

While the inequality holds for non-negative numbers, the geometric mean is most commonly applied and defined for positive numbers, as taking roots of negative numbers (especially even roots) can lead to complex numbers or undefined results within the real number system.

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Proof for Two Positive Numbers

Let's demonstrate the AM-GM inequality for the simplest case, involving just two positive real numbers, $a$ and $b$. We aim to prove that $\frac{a+b}{2} \ge \sqrt{ab}$.

We can start by considering a fundamental property of real numbers: the square of any real number is always non-negative (greater than or equal to zero). Since $a$ and $b$ are positive, their square roots $\sqrt{a}$ and $\sqrt{b}$ are real numbers.

Consider the difference between $\sqrt{a}$ and $\sqrt{b}$, and square it:

$$ (\sqrt{a} - \sqrt{b})^2 \ge 0 $$

Now, expand the left side using the algebraic identity $(x-y)^2 = x^2 - 2xy + y^2$. Here, $x=\sqrt{a}$ and $y=\sqrt{b}$:

$$ (\sqrt{a})^2 - 2(\sqrt{a})(\sqrt{b}) + (\sqrt{b})^2 \ge 0 $$

Simplify the terms ($\left(\sqrt{a}\right)^2 = a$, $\left(\sqrt{b}\right)^2 = b$, and $\sqrt{a}\sqrt{b} = \sqrt{ab}$ since $a,b \ge 0$):

$$ a - 2\sqrt{ab} + b \ge 0 $$

Rearrange the terms by adding $2\sqrt{ab}$ to both sides of the inequality:

$$ a + b \ge 2\sqrt{ab} $$

Finally, divide both sides by $2$. Since $2$ is a positive number, the direction of the inequality sign does not change:

$$ \frac{a+b}{2} \ge \sqrt{ab} $$

This completes the proof of the AM-GM inequality for two positive numbers.

Condition for Equality:

The equality in the inequality $\frac{a+b}{2} \ge \sqrt{ab}$ holds precisely when the original squared term $(\sqrt{a} - \sqrt{b})^2$ is equal to zero.

$$ (\sqrt{a} - \sqrt{b})^2 = 0 $$

Taking the square root of both sides:

$$ \sqrt{a} - \sqrt{b} = 0 $$ $$ \sqrt{a} = \sqrt{b} $$

Since $a$ and $b$ are positive numbers, squaring both sides gives:

$$ a = b $$

Therefore, the equality $\frac{a+b}{2} = \sqrt{ab}$ holds if and only if $a=b$. If $a \neq b$, then $(\sqrt{a} - \sqrt{b})^2$ is strictly greater than 0, leading to $\frac{a+b}{2} > \sqrt{ab}$.

The proof for the general case of $n$ numbers is more involved and often uses mathematical induction or Jensen's inequality.

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Significance and Applications of the AM-GM Inequality

The AM-GM inequality is a powerful tool with wide applications in various branches of mathematics, including algebra, calculus, and optimization. It is frequently used to:

• Find Minimum or Maximum Values: The inequality can help in finding the least possible value of a sum (when the product is fixed) or the greatest possible value of a product (when the sum is fixed).
• Prove Other Inequalities: Many other inequalities can be derived as consequences of the AM-GM inequality.
• Solve Optimization Problems: It is useful in solving certain types of optimization problems where we need to find the extreme values of expressions involving positive quantities.

Example Application: Find the minimum value of the expression $x + \frac{1}{x}$ for positive real numbers $x > 0$.

We can apply the AM-GM inequality to the two positive numbers $x$ and $\frac{1}{x}$. According to the AM-GM inequality:

$$ \frac{x + \frac{1}{x}}{2} \ge \sqrt{x \times \frac{1}{x}} $$

Simplify the expression under the square root:

$$ x \times \frac{1}{x} = 1 $$

So the inequality becomes:

$$ \frac{x + \frac{1}{x}}{2} \ge \sqrt{1} $$ $$ \frac{x + \frac{1}{x}}{2} \ge 1 $$

Multiply both sides by 2:

$$ x + \frac{1}{x} \ge 2 $$

This inequality tells us that the value of $x + \frac{1}{x}$ is always greater than or equal to 2 for any positive $x$. Therefore, the minimum value of the expression $x + \frac{1}{x}$ is 2.

When is this minimum value achieved? The equality in the AM-GM inequality holds if and only if the two numbers are equal. In this case, the two numbers are $x$ and $\frac{1}{x}$.

$$ x = \frac{1}{x} $$

Multiply both sides by $x$ (since $x > 0$, $x \neq 0$):

$$ x^2 = 1 $$

Taking the square root, $x = \pm 1$. Since we are given $x > 0$, the minimum value occurs when $x = 1$.

Check: When $x=1$, $1 + \frac{1}{1} = 1 + 1 = 2$. This confirms the minimum value.

The AM-GM inequality is a fundamental result that highlights the difference between arithmetic and geometric averages and is a versatile tool for solving a variety of mathematical problems.

Some Special Series (Summation techniques)

While Arithmetic Progressions (APs) and Geometric Progressions (GPs) are the most common types of sequences and series encountered in introductory algebra, there are many other sequences and series that follow different patterns. Finding the sum of the first $n$ terms of these "special" series often requires different summation techniques or relies on specific formulas derived for the sums of powers of natural numbers.

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Sum of the First n Terms of Some Special Series

There are standard formulas for the sum of the first $n$ terms of several important series involving the positive integers (natural numbers):

1. Sum of the First n Positive Integers

This is the series formed by adding the first $n$ natural numbers: $1 + 2 + 3 + \ldots + n$. As we saw in the section on the sum of an AP, this specific series is actually an Arithmetic Progression with the first term $a=1$ and the common difference $d=1$. The sum of its first $n$ terms, $S_n$, can be calculated using the AP sum formula $S_n = \frac{n}{2}(a+l)$ where $l=n$, or $S_n = \frac{n}{2}[2a+(n-1)d]$.

Using $a=1$ and $l=n$ in $S_n = \frac{n}{2}(a+l)$ gives:

$$ S_n = \frac{n}{2}(1 + n) $$ $$ S_n = \frac{n(n+1)}{2} $$

Derivation (using Pairing Method):

Let $S_n = 1 + 2 + \ldots + (n-1) + n$. Write the sum again in reverse order: $S_n = n + (n-1) + \ldots + 2 + 1$.

Add the two sums vertically:

$$ S_n = 1 \quad + \quad 2 \quad + \quad \ldots \quad + \quad (n-1) \quad + \quad n $$ $$ S_n = n \quad + \quad (n-1) \quad + \quad \ldots \quad + \quad 2 \quad + \quad 1 $$ $$ \rule{5cm}{0.4pt} $$ $$ 2S_n = (1+n) + (2+(n-1)) + \ldots + ((n-1)+2) + (n+1) $$ $$ 2S_n = (n+1) + (n+1) + \ldots + (n+1) + (n+1) $$

There are $n$ terms in the sum, so there are $n$ pairs, each summing to $(n+1)$.

$$ 2S_n = n(n+1) $$

Divide by 2:

$$ S_n = \frac{n(n+1)}{2} $$

Example: Find the sum of the first 5 positive integers.

Using direct summation: $1 + 2 + 3 + 4 + 5 = 15$.

Using the formula with $n=5$: $S_5 = \frac{5(5+1)}{2} = \frac{5 \times 6}{2} = \frac{30}{2} = 15$. The formula works.

2. Sum of the Squares of the First n Positive Integers

This series consists of the sum of the squares of the first $n$ natural numbers: $1^2 + 2^2 + 3^2 + \ldots + n^2$. This sequence of terms ($1, 4, 9, \ldots$) is neither an AP nor a GP. There is a specific formula for its sum, $\sum_{i=1}^{n} i^2$:

Derivation (Outline using a Telescoping Sum):

Consider the identity $(k+1)^3 - k^3 = 3k^2 + 3k + 1$. Write this identity for $k=1, 2, \ldots, n$:

• $k=1$: $2^3 - 1^3 = 3(1)^2 + 3(1) + 1$
• $k=2$: $3^3 - 2^3 = 3(2)^2 + 3(2) + 1$
• $k=3$: $4^3 - 3^3 = 3(3)^2 + 3(3) + 1$
• ...
• $k=n$: $(n+1)^3 - n^3 = 3(n)^2 + 3(n) + 1$

Summing all these equations vertically, notice the left side is a telescoping sum where intermediate terms cancel out:

$$ (2^3 - 1^3) + (3^3 - 2^3) + (4^3 - 3^3) + \ldots + ((n+1)^3 - n^3) = (n+1)^3 - 1^3 $$

The sum of the right side is:

$$ \sum_{k=1}^{n} (3k^2 + 3k + 1) = 3 \sum_{k=1}^{n} k^2 + 3 \sum_{k=1}^{n} k + \sum_{k=1}^{n} 1 $$

Let $S_n^{(2)} = \sum_{k=1}^{n} k^2$. We know $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$ and $\sum_{k=1}^{n} 1 = n$.

So, $(n+1)^3 - 1 = 3 S_n^{(2)} + 3 \frac{n(n+1)}{2} + n$.

Expand $(n+1)^3 = n^3 + 3n^2 + 3n + 1$.

$n^3 + 3n^2 + 3n + 1 - 1 = 3 S_n^{(2)} + \frac{3n(n+1)}{2} + n$

$n^3 + 3n^2 + 3n = 3 S_n^{(2)} + \frac{3n^2 + 3n}{2} + n$

$3 S_n^{(2)} = n^3 + 3n^2 + 3n - \frac{3n^2 + 3n}{2} - n$

$3 S_n^{(2)} = n^3 + 3n^2 + 2n - \frac{3n^2 + 3n}{2}$

$3 S_n^{(2)} = \frac{2n^3 + 6n^2 + 4n - (3n^2 + 3n)}{2}$

$3 S_n^{(2)} = \frac{2n^3 + 3n^2 + n}{2}$

$3 S_n^{(2)} = \frac{n(2n^2 + 3n + 1)}{2}$

Factor the quadratic $2n^2 + 3n + 1 = (2n+1)(n+1)$.

$3 S_n^{(2)} = \frac{n(n+1)(2n+1)}{2}$

Divide by 3:

$$ S_n^{(2)} = \frac{n(n+1)(2n+1)}{6} $$

Example: Find the sum of the squares of the first 3 positive integers.

Using direct summation: $1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14$.

Using the formula with $n=3$: $S_3 = \frac{3(3+1)(2 \times 3 + 1)}{6} = \frac{3 \times 4 \times (6+1)}{6} = \frac{3 \times 4 \times 7}{6} = \frac{84}{6} = 14$. The formula matches.

3. Sum of the Cubes of the First n Positive Integers

This series is the sum of the cubes of the first $n$ natural numbers: $1^3 + 2^3 + 3^3 + \ldots + n^3$. This is also not an AP or a GP. The formula for this sum, $\sum_{i=1}^{n} i^3$, has a remarkable relationship with the sum of the first $n$ integers:

This formula states that the sum of the first $n$ cubes is equal to the square of the sum of the first $n$ natural numbers.

Derivation (Outline using a Telescoping Sum):

Consider the identity $(k+1)^4 - k^4 = 4k^3 + 6k^2 + 4k + 1$. Summing this identity for $k=1, 2, \ldots, n$ gives a telescoping sum on the left side: $(n+1)^4 - 1^4$.

Summing the right side: $\sum_{k=1}^{n} (4k^3 + 6k^2 + 4k + 1) = 4 \sum k^3 + 6 \sum k^2 + 4 \sum k + \sum 1$.

Substitute the known formulas for $\sum k^2$, $\sum k$, and $\sum 1$, and solve for $\sum k^3$. This leads to the formula $\left(\frac{n(n+1)}{2}\right)^2$. (This derivation is algebraically more intensive than the one for squares).

Example: Find the sum of the cubes of the first 3 positive integers.

Using direct summation: $1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36$.

Using the formula with $n=3$: $S_3 = \left(\frac{3(3+1)}{2}\right)^2 = \left(\frac{3 \times 4}{2}\right)^2 = \left(\frac{12}{2}\right)^2 = (6)^2 = 36$. The formula holds.

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Other Summation Techniques

For sequences and series that do not fit the patterns of AP, GP, or simple sums of powers, various other techniques are employed to find the sum of a finite number of terms or to determine the behavior (convergence or divergence) of an infinite series. Some common methods include:

• Method of Differences (Telescoping Sums): This technique is applicable when the general term $a_n$ of the series can be expressed as the difference of two consecutive terms of another sequence, say $b_n$, such that $a_n = b_n - b_{n-1}$. When the sum $\sum_{i=1}^n a_i$ is expanded, most intermediate terms cancel out, leaving only the first and last terms of the sequence $b_n$: $$ \sum_{i=1}^n a_i = \sum_{i=1}^n (b_i - b_{i-1}) = (b_1 - b_0) + (b_2 - b_1) + (b_3 - b_2) + \ldots + (b_n - b_{n-1}) $$ $$ = b_n - b_0 $$

  Example: Consider the series $\sum_{i=1}^n \frac{1}{i(i+1)}$. The general term $a_i = \frac{1}{i(i+1)}$ can be decomposed using partial fractions: $\frac{1}{i(i+1)} = \frac{1}{i} - \frac{1}{i+1}$. Let $b_i = \frac{-1}{i+1}$ and $b_{i-1} = \frac{-1}{i}$. Then $b_i - b_{i-1} = \frac{-1}{i+1} - (\frac{-1}{i}) = \frac{1}{i} - \frac{1}{i+1}$. No, let $b_i = \frac{1}{i}$. Then $a_i = \frac{1}{i(i+1)}$. We want $a_i = b_i - b_{i+1}$ or similar. Let's try $a_i = c_i - c_{i+1}$. $\frac{1}{i(i+1)} = \frac{1}{i} - \frac{1}{i+1}$. Let $c_i = \frac{1}{i}$. Then $c_{i+1} = \frac{1}{i+1}$. So $a_i = c_i - c_{i+1}$.

  The sum is $\sum_{i=1}^n \left(\frac{1}{i} - \frac{1}{i+1}\right) = \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \ldots + \left(\frac{1}{n} - \frac{1}{n+1}\right)$.

  All intermediate terms cancel out, leaving $1 - \frac{1}{n+1} = \frac{n+1-1}{n+1} = \frac{n}{n+1}$. The sum of the first $n$ terms is $\frac{n}{n+1}$.

• Using Algebraic Identities: Some sums can be found by recognizing connections to algebraic identities or expansions.
• Recognizing Patterns in Partial Sums: By calculating the sums of the first few terms ($S_1, S_2, S_3, \ldots$), a pattern might emerge that suggests a formula for $S_n$. This formula can then be verified (e.g., using mathematical induction).

The study of series extends significantly in higher mathematics, including topics like convergence and divergence of infinite series, power series expansions of functions, and Fourier series.

Applications of AP and GP

Arithmetic Progressions (APs) and Geometric Progressions (GPs) are not just abstract mathematical concepts confined to textbooks. They are powerful tools that can be used to model and solve problems involving sequences where quantities change in a regular, predictable manner. These patterns appear in various real-world situations, from finance and economics to physics and biology.

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Applications of Arithmetic Progressions (AP)

An Arithmetic Progression is characterized by a constant difference between successive terms. This makes APs suitable for modeling situations where a quantity increases or decreases by a fixed *additive* amount over equal intervals.

Key Applications of APs:

1. Linear Growth or Decay: Any scenario where a quantity changes by a fixed amount in each step or period can be modeled by an AP.

   Example: If a taxi charges a base fare and a fixed amount per kilometre, the total fare increases arithmetically with the distance traveled. The costs for 1 km, 2 km, 3 km, etc., form an AP.

   Example: The depreciation of an asset by a fixed amount each year (straight-line depreciation). The value of the asset at the end of each year forms an AP.

2. Sequences of Events or Values at Regular Intervals: Problems involving items or events spaced out by a constant difference.

   Example: The numbers on the seats in a row in a theatre if they are numbered consecutively (e.g., Row A: 101, 102, 103, ...).

   Example: The distances of equally spaced trees planted along a straight road from a starting point.

3. Physical Structures: Designs involving dimensions that change uniformly.

   Example: The lengths of ladder rungs that decrease uniformly from the bottom to the top form an AP. To find the total length of wood required, we find the sum of this AP.

4. Salaries and Wages with Fixed Increments: If an employee receives an annual salary increment of a fixed amount, their salaries over the years form an AP.

   Example: Starting salary $\textsf{₹} 20,000$ with an annual increment of $\textsf{₹} 1000$. Salaries are $\textsf{₹} 20,000, \textsf{₹} 21,000, \textsf{₹} 22,000, \ldots$

5. Simple Interest: While the total amount (Principal + Simple Interest) grows arithmetically, the amount of simple interest earned each period remains constant. The sequence of total amounts after each period under simple interest forms an AP. If principal is $P$, rate is $R\%$ per annum, the total amount after $n$ years is $P + \frac{PRn}{100} = P + (\frac{PR}{100})n$. This is a linear function of $n$. The amounts at the end of year 0, 1, 2, ... are $P, P + \frac{PR}{100}, P + \frac{2PR}{100}, \ldots$. This is an AP with first term $P$ and common difference $\frac{PR}{100}$.

Example Application of AP

Answer:

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Applications of Geometric Progressions (GP)

A Geometric Progression is characterized by a constant ratio between successive terms. This makes GPs suitable for modeling situations where a quantity changes by a fixed *multiplicative* factor (or a constant percentage rate) over equal intervals. This type of change is often referred to as exponential growth or decay.

Key Applications of GPs:

1. Exponential Growth and Decay: Many natural and financial phenomena exhibit growth or decay that is proportional to the current amount.

   Example: Compound Interest. If a principal amount $P$ is invested at an interest rate $R$ per period compounded periodically, the amount at the end of each period increases by a factor of $(1+R)$. The amounts at the end of periods 1, 2, 3, ..., form a GP: $P(1+R), P(1+R)^2, P(1+R)^3, \ldots$. This is a GP with first term $a = P(1+R)$ and common ratio $r = (1+R)$. Alternatively, the sequence of amounts at the *start* of each period $0, 1, 2, 3, \ldots$ years forms a GP: $P, P(1+R), P(1+R)^2, P(1+R)^3, \ldots$ with first term $a=P$ and common ratio $r=(1+R)$. The amount after $n$ periods is the $(n+1)$-th term of this sequence, which is $a_{n+1} = P(1+R)^{(n+1)-1} = P(1+R)^n$.

   Example: Population Growth. If a population increases by a fixed percentage each year, the population figures for consecutive years may approximate a GP.

   Example: Radioactive Decay. The amount of a radioactive substance remaining after successive equal time intervals decreases by a constant ratio.

   Example: Depreciation at a fixed percentage rate each year (e.g., value of a car or machinery). The value at the end of each year forms a GP.

2. Chain Reactions: Processes where each event triggers a fixed number of subsequent events (e.g., nuclear fission chain reactions, certain biological reproduction processes where one organism produces a fixed number of offspring).
3. Successive Bounces: The maximum height reached by a bouncing ball after each bounce often forms a GP, assuming a consistent loss of energy (by a fixed ratio) with each impact.
4. Spread of Information or Disease: In simple models, the number of people newly informed or infected in successive time periods might follow a geometric pattern if each existing case generates a fixed number of new cases.

Example Application of GP

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In summary, APs and GPs are powerful mathematical models for situations involving linear and exponential patterns of change, respectively. They provide efficient ways to calculate future values, sums, or other properties of sequences that follow these predictable patterns, making them valuable tools in various quantitative disciplines.