Chapter 9 Sequences and Series (Concepts)

Sequence

The concept of sequences plays a vital role in various fields of human activity, ranging from simple counting to complex financial modeling and biological research. In everyday life, whenever we arrange a collection of objects in a specific order such that we can identify a first, second, and third member, we are essentially forming a sequence.

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Real-Life Applications of Sequences

Sequences are not just abstract mathematical concepts; they are reflected in practical scenarios:

1. Banking and Finance: The amount of money accumulated in a fixed deposit (FD) in an Indian bank over a period of years forms a sequence based on the interest rate.

2. Economics: The depreciated value of commodities like machinery or vehicles over time follows a specific sequence.

3. Biology: The growth of a bacteria population at fixed time intervals can be represented as a sequence.

4. Human Ancestry: Consider the number of ancestors a person has over 10 generations (parents, grandparents, great-grandparents, etc.). The number of ancestors for the first, second, and third generations would be 2, 4, and 8 respectively. For 10 generations, the sequence ends at 1024.

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Formal Definition of a Sequence

A set of numbers arranged in a definite order according to a definite rule (or set of rules) is called a sequence. Each number belonging to this set is referred to as a term of the sequence.

Notation and Terminology

• The terms are usually denoted by $a_1, a_2, a_3, \dots, a_n$ or $T_1, T_2, T_3, \dots, T_n$.

• The subscript represents the position of the term in the sequence and must always be a natural number ($n \in \mathbb{N}$).

• The term occurring at the $n^{th}$ position, denoted by $a_n$, is known as the General Term of the sequence.

Classification of Sequences

1. Finite Sequence: A sequence containing a fixed number of terms. It is described as $a_1, a_2, a_3, \dots, a_n$.

2. Infinite Sequence: A sequence that continues indefinitely. It is described as $a_1, a_2, a_3, \dots$ to $\infty$.

3. Real Sequence: If every term of the sequence is a real number, it is called a real sequence.

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Analysis of Patterns and Rules

To define a sequence, we must identify the rule governing its terms. A sequence is not merely a random collection of numbers but a structured list where each term follows a specific logical or mathematical relationship with its position or its preceding terms.

The rules governing sequences are generally categorized into Recursive Rules (where a term depends on the previous term) and Explicit Rules (where a term depends directly on its position $n$).

Detailed Classification of Sequence Rules

Let us analyze the mathematical structure of the common patterns observed in sequences through the following vertical classification:

Sequence Type	Example Pattern	Mathematical Rule ($a_n$)
Additive (Arithmetic)	$3, 5, 7, 9, \dots$	$a_n = a_{n-1} + 2$
Subtractive	$8, 5, 2, -1, \dots$	$a_n = a_{n-1} - 3$
Multiplicative (Geometric)	$2, 6, 18, 54, \dots$	$a_n = 3 \cdot a_{n-1}$
Power-Based (Squares)	$1, 4, 9, 16, \dots$	$a_n = n^2$
Fractional / Reciprocal	$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$	$a_n = \frac{1}{n}$
Alternating Signs	$-1, 2, -3, 4, \dots$	$a_n = (-1)^n \cdot n$

Derivation of the General Term ($a_n$)

The process of finding the General Term involves observing the relationship between the term value and its position $n$.

1. Case of Constant Addition (Linear Pattern)

Consider the sequence: $3, 5, 7, 9, \dots$

By induction, the general term for the $n^{th}$ position is:

2. Case of Constant Multiplication (Exponential Pattern)

Consider the sequence: $2, 6, 18, 54, \dots$

The power of 3 is always $(n-1)$. Thus, the general term is:

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Progressions

In mathematical analysis, a Progression is a specialized type of sequence where the terms follow a very specific and predictable pattern that can be expressed via an explicit algebraic formula. This formula allows for the direct calculation of any $n^{th}$ term without necessarily knowing the preceding terms.

While the terms "sequence" and "progression" are often used interchangeably in casual conversation, a technical distinction exists. Every progression is a sequence, but not every sequence qualifies as a progression. To be a progression, the sequence must exhibit a uniform mathematical regularity.

Comparative Analysis: Sequence vs. Progression

The following table distinguishes between general sequences and progressions based on their mathematical properties:

Feature	General Sequence	Progression
Rule	May be descriptive or logical.	Must be a definite mathematical formula.
Explicit Formula	Not always available.	Always exists for the $n^{th}$ term.
Predictability	Difficult for large values of $n$ (if no formula).	Easily predictable for any $n$.
Example	Prime Numbers ($2, 3, 5, \dots$)	Arithmetic Progression ($2, 4, 6, \dots$)

The Case of Prime Numbers

The set of Prime Numbers $\{2, 3, 5, 7, 11, 13, \dots\}$ is an excellent example of a sequence that is not a progression. Although there is a clear logical rule (a prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself), there is no known simple algebraic expression $f(n)$ that generates the $n^{th}$ prime number. Therefore, it remains a sequence but fails the criteria for a progression.

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The Fibonacci Sequence (Matrameru)

The Fibonacci Sequence is one of the most significant progressions in mathematics. Historically, this pattern was described in Indian mathematics by Pingala in connection with Sanskrit prosody (the study of poetic meters) as Matrameru, long before it was known in the West.

In this progression, each term is the sum of the two preceding terms. It is defined by a Recurrence Relation:

Where the initial terms are defined as:

Derivation of Fibonacci Terms

To find the first few terms, we apply the relation systematically:

• Third term ($a_3$):

• Fourth term ($a_4$):

• Fifth term ($a_5$):

The resulting progression is: $1, 1, 2, 3, 5, 8, 13, 21, \dots$

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Series

The concept of a series arises when the individual terms of a sequence are connected by addition signs. While a sequence is simply a list of numbers, a series represents the indicated sum of those numbers. If we consider a sequence $a_1, a_2, a_3, \dots, a_n$, then the expression formed by their sum is known as the series associated with that sequence.

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Formal Definition

Let $a_1, a_2, a_3, \dots$ be a given sequence. Then the expression:

$a_1 + a_2 + a_3 + \dots$

is called the series associated with the sequence.

Like sequences, series are also classified into two types:

1. Finite Series: A series derived from a finite sequence, having a specific number of terms. Example: $3 + 5 + 7 + \dots + 21$.

2. Infinite Series: A series derived from an infinite sequence, continuing without end. Example: $1 + 4 + 9 + 16 + \dots$ to $\infty$.

Sigma Notation and Summation

To represent a series compactly, mathematicians use the Greek symbol Sigma ($\Sigma$). If $a_k$ denotes the general term of a sequence, the sum of the first $n$ terms is denoted by $S_n$ and is written as:

Important Note: In mathematical terminology, the word 'series' refers to the indicated sum (the expression) rather than the numerical result. The numerical result is specifically called the 'sum of the series'. For instance, $1 + 3 + 5$ is a series, whereas $9$ is its sum.

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Relationship between $a_n$ and $S_n$

There is a fundamental relationship between the general term of a sequence ($a_n$) and the sum of its terms ($S_n$). The $n^{th}$ term can be found by subtracting the sum of the first $(n-1)$ terms from the sum of the first $n$ terms.

Derivation

The sum of $n$ terms is:

The sum of $(n-1)$ terms is:

Subtracting $S_{n-1}$ from $S_n$:

All terms except $a_n$ cancel out, leading to the identity:

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Examples of Series Patterns

Based on the examples of sequences, we can form various series by connecting the terms with plus signs:

Sequence Type	Corresponding Series	Nature
Arithmetic	$3 + 5 + 7 + 9 + \dots + 21$	Finite
Geometric	$2 + 6 + 18 + 54 + \dots + 1458$	Finite
Squares	$1 + 4 + 9 + 16 + \dots$	Infinite
Alternating	$1 + (-1) + 1 + (-1) + \dots$	Infinite

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Arithmetic Progression

An Arithmetic Progression (A.P.) is a sequence in which the difference between any term and its preceding term is always a constant. This constant is called the common difference of the A.P. and is typically denoted by the letter $d$. If the sequence is finite, the indicated sum of its terms is called an arithmetic series.

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Formal Definition and the Nature of Common Difference

A sequence $a_1, a_2, a_3, \dots, a_n$ is called an Arithmetic Progression if the difference between any term and its immediate predecessor is a constant. This constant value is known as the Common Difference, denoted by $d$.

The behavior of the A.P. is entirely determined by the sign of $d$:

Condition	Type of A.P.	Example Sequence
$d > 0$	Increasing A.P.	$5, 10, 15, 20, \dots$ ($d = 5$)
$d < 0$	Decreasing A.P.	$100, 90, 80, 70, \dots$ ($d = -10$)
$d = 0$	Constant A.P.	$\textsf{₹} 50, \textsf{₹} 50, \textsf{₹} 50, \dots$ ($d = 0$)

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The General Term ($n^{th}$ Term)

To find a formula that gives the value of the $n^{th}$ term directly, let us denote the first term as $a$ and the common difference as $d$. We build the sequence term by term:

1. The first term ($a_1$):

2. The second term ($a_2$):

3. The third term ($a_3$):

4. The fourth term ($a_4$):

By observing the subscript of the term and the coefficient of $d$, we notice that the coefficient of $d$ is always one less than the position of the term. Thus, for the $n^{th}$ term:

If a finite A.P. has $n$ terms, the last term $l$ is identical to $a_n$:

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$n^{th}$ Term from the End

Calculating a term starting from the end of a finite sequence is a common requirement in exams. Suppose we have a finite A.P. with a total of $m$ terms.

Method 1: Transformation to Beginning Index

The $n^{th}$ term from the end is not simply $a_{m-n}$. To visualize this, consider a sequence of 5 terms: $T_1, T_2, T_3, T_4, T_5$.

• $1^{st}$ from end = $5^{th}$ from start $= (5 - 1 + 1)^{th}$ term.

• $2^{nd}$ from end = $4^{th}$ from start $= (5 - 2 + 1)^{th}$ term.

Thus, the $n^{th}$ term from the end is the $(m - n + 1)^{th}$ term from the beginning.

Method 2: Reversing the Sequence

Alternatively, we can treat the last term ($l$) as our new first term. When we move backwards through the A.P., the common difference changes sign (from $d$ to $-d$).

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Sum of the First $n$ Terms ($S_n$)

The sum of the first $n$ terms of an A.P. is the value obtained by adding all terms from $a_1$ to $a_n$. This is particularly useful in financial calculations, such as determining the total interest paid over a loan tenure.

Derivation of the Sum Formula

Let $a$ be the first term, $d$ the common difference, $n$ the number of terms, and $l$ the last term. We can write the sum $S_n$ in two ways:

1. Writing the sum in forward order:

2. Writing the same sum in reverse order:

By adding equations (i) and (ii) vertically, we observe that each pair of terms sums to $(a + l)$:

This gives us the first form of the sum formula:

Since the last term $l = a + (n - 1)d$, substituting this in the above equation gives us the second form:

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Algebraic Properties of Arithmetic Progressions

A.P. sequences exhibit remarkable stability under basic arithmetic operations. These properties are frequently used to transform complex sequences into simpler forms.

Linear Transformations

If every term of an A.P. is modified by a constant $k$, the resulting sequence remains an A.P.

Operation	Original AP	Modified AP	New Common Difference
Addition	$a, a+d, a+2d$	$a+k, (a+k)+d, (a+k)+2d$	$d$
Subtraction	$a, a+d, a+2d$	$a-k, (a-k)+d, (a-k)+2d$	$d$
Multiplication	$a, a+d, a+2d$	$ak, ak+dk, ak+2dk$	$dk$
Division ($k \neq 0$)	$a, a+d, a+2d$	$\frac{a}{k}, \frac{a+d}{k}, \frac{a+2d}{k}$	$\frac{d}{k}$

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Selection of Terms for Problem Solving

In many algebraic problems where the sum of terms is provided, selecting specific symmetric terms can eliminate the variable $d$, making it easier to find the value of $a$.

Number of Terms	Recommended Selection	Common Difference
3 Terms	$a - d, \ a, \ a + d$	$d$
4 Terms	$a - 3d, \ a - d, \ a + d, \ a + 3d$	$2d$
5 Terms	$a - 2d, \ a - d, \ a, \ a + d, \ a + 2d$	$d$

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Arithmetic Mean and Symmetry

1. Arithmetic Mean (A.M.)

If three numbers $a, b,$ and $c$ are in A.P., then $b$ is the Arithmetic Mean of $a$ and $c$.

2. Property of Equidistant Terms

In any finite A.P., the sum of the terms equidistant from the beginning and the end is always constant and equal to the sum of the first and last terms.

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Arithmetic Mean

The concept of Arithmetic Mean (A.M.) is fundamental in sequences. When one or more numbers are inserted between two given numbers such that the resulting sequence forms an Arithmetic Progression, these inserted numbers are known as Arithmetic Means. In simple terms, for any two numbers, the A.M. is their average, but in progressions, it serves as the bridge that maintains a constant common difference.

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Arithmetic Mean Between Two Numbers

Given two numbers $a$ and $b$, we can insert a number $A$ between them such that $a, A, b$ form an A.P. This number $A$ is the single Arithmetic Mean of $a$ and $b$.

Derivation of the Single A.M.

Since the sequence $a, A, b$ is in A.P., the difference between consecutive terms must be equal to the common difference $d$.

Rearranging the terms to solve for $A$:

Thus, the single A.M. of two numbers is their mathematical average.

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Inserting $n$ Arithmetic Means Between Two Numbers

We can insert $n$ numbers, say $A_1, A_2, A_3, \dots, A_n$, between two given numbers $a$ and $b$ such that the sequence $a, A_1, A_2, A_3, \dots, A_n, b$ forms an A.P.

Derivation of Common Difference ($d$)

In the resulting A.P., we have:

• The first term = $a$

• The last term = $b$

• Total number of terms = $n$ (inserted means) $+ 2$ (original numbers) $= n + 2$

Let $d$ be the common difference. Since $b$ is the $(n + 2)^{th}$ term:

Values of Individual Means

Once $d$ is found, the individual means are calculated as follows:

$\vdots$

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Sum of $n$ Arithmetic Means

An important property is that the sum of $n$ arithmetic means inserted between $a$ and $b$ is always equal to $n$ times the single A.M. between $a$ and $b$.

Proof:

The sum of $n$ means is given by:

$S = A_1 + A_2 + \dots + A_n$

Since the means themselves form an A.P. with $n$ terms, first term $A_1 = (a + d)$ and last term $A_n = (b - d)$:

$S = \frac{n}{2}(A_1 + A_n)$

$S = \frac{n}{2}(a + d + b - d)$

$S = \frac{n}{2}(a + b) = n \left( \frac{a + b}{2} \right)$

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General Remarks on Sum of A.P.

If an A.P. consists of $N$ terms, its sum can be expressed in relation to its middle terms:

1. When $N$ is odd: The sum of the $N$ terms is equal to $N$ times the middle term.

2. When $N$ is even: The sum of the $N$ terms is equal to $N$ times the arithmetic mean of the two middle terms.

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Geometric Progression

A Geometric Progression (G.P.) is a sequence of non-zero numbers where each term after the first is obtained by multiplying the preceding term by a fixed, non-zero constant. This constant is known as the common ratio, typically denoted by $r$. In a G.P., the ratio of any term to its immediate predecessor remains invariant throughout the sequence.

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Definition and Nature of Common Ratio

A sequence of non-zero numbers $a_1, a_2, a_3, \dots, a_n$ is a G.P. if the ratio of any term to its preceding term is a constant $r$.

The behavior of a G.P. depends on the value of $r$:

• Increasing G.P.: If $a > 0$ and $r > 1$, the terms increase rapidly (e.g., $2, 4, 8, 16, \dots$).

• Decreasing G.P.: If $a > 0$ and $0 < r < 1$, the terms decrease towards zero (e.g., $100, 50, 25, 12.5, \dots$).

• Alternating G.P.: If $r < 0$, the signs of the terms alternate between positive and negative (e.g., $3, -6, 12, -24, \dots$).

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Derivation of the General Term ($n^{th}$ Term)

Let $a$ be the first term and $r$ be the common ratio. The sequence is constructed as follows:

Observing the exponent of $r$, we see it is always one less than the position of the term $n$. Thus, the General Term is:

For a finite G.P. with $n$ terms, the last term $l$ is:

Finding the $n^{th}$ Term from the End

In competitive exams, you may be asked to find a term starting from the end of a finite G.P. having $m$ terms.

Method 1: Index Transformation

The $n^{th}$ term from the end corresponds to the $(m - n + 1)^{th}$ term from the beginning.

Method 2: Reversing the Sequence

If we look at the G.P. from the end, the last term $l$ becomes the first term, and the common ratio becomes its reciprocal $\frac{1}{r}$.

Strategic Selection of Terms in G.P.

When solving problems where the product of terms is given, choosing symmetric terms allows $r$ to cancel out during multiplication, simplifying the calculation of $a$.

Number of Terms	Recommended Selection	Common Ratio
3 Terms	$\frac{a}{r}, \ a, \ ar$	$r$
4 Terms	$\frac{a}{r^3}, \ \frac{a}{r}, \ ar, \ ar^3$	$r^2$
5 Terms	$\frac{a}{r^2}, \ \frac{a}{r}, \ a, \ ar, \ ar^2$	$r$

Logic: If the product of 3 terms is $K$, then $(\frac{a}{r})(a)(ar) = K \implies a^3 = K \ $$ \implies a = \sqrt[3]{K}$.

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Fundamental Property of Three Terms

Three non-zero numbers $a, b, c$ are in G.P. if and only if the ratio between $b$ and $a$ is the same as the ratio between $c$ and $b$.

Here, $b$ is known as the Geometric Mean of $a$ and $c$, expressed as $b = \pm \sqrt{ac}$.

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Derivation of the Finite Sum Formula ($S_n$)

Let $a$ be the first term, $r$ the common ratio, and $n$ the total number of terms. The sum of the first $n$ terms is denoted by $S_n$.

To derive the formula, we multiply equation (i) by the common ratio $r$:

Subtracting equation (ii) from equation (i):

$S_n - rS_n = (a + ar + ar^2 + \dots + ar^{n-1}) - (ar + ar^2 + \dots + ar^n)$

Upon subtraction, all the middle terms cancel out, leaving only the first term of the first sequence and the last term of the second sequence:

$S_n(1 - r) = a - ar^n$

$S_n(1 - r) = a(1 - r^n)$

Dividing by $(1 - r)$, we obtain the primary summation formulas:

1. Formula for $|r| < 1$

This version is preferred when the ratio is a fraction less than 1 to keep the denominator positive:

2. Formula for $|r| > 1$

This version is used when the terms are growing in magnitude:

Note: If $r = 1$, the formulas (iii) and (iv) result in division by zero. In this case, $S_n = a + a + \dots$ ($n$ times), so:

3. Relation with Last Term ($l$)

If the last term $l = ar^{n-1}$ is known, the sum can be expressed without explicitly calculating $n$:

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Sum of an Infinite Geometric Progression ($S_\infty$)

For a geometric series where the number of terms is infinite, a finite sum exists only if the magnitude of the common ratio is less than unity ($|r| < 1$).

Derivation of the Infinite Sum

As the number of terms $n$ approaches infinity, we analyze the behavior of the term $r^n$ in formula (iii):

Substituting this limit into the finite sum formula:

$S_\infty = \frac{a(1 - 0)}{1 - r}$

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Geometric Mean

The Geometric Mean (G.M.) is a specific type of average that indicates the central tendency of a set of numbers by using the product of their values. In the context of progressions, if a number is inserted between two given numbers such that the resulting trio forms a Geometric Progression, that number is defined as the Geometric Mean.

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Geometric Mean Between Two Positive Numbers

Given two different positive numbers $a$ and $b$, we can insert a number $G$ between them such that the sequence $a, G, b$ forms a Geometric Progression (G.P.). This number $G$ is called the geometric mean of $a$ and $b$.

Derivation of the Formula

If $a, G, b$ are in G.P., then by the definition of common ratio ($r$):

By cross-multiplication, we obtain:

Taking the square root of both sides:

Hence, the geometric mean of two positive numbers is the positive square root of their product.

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Inserting $n$ Geometric Means Between Two Numbers

We can insert multiple numbers $G_1, G_2, G_3, \dots, G_n$ between two positive numbers $a$ and $b$ such that the resulting sequence $a, G_1, G_2, G_3, \dots, G_n, b$ is a G.P. These $n$ numbers are called the $n$ geometric means between $a$ and $b$.

Derivation of Common Ratio ($r$)

In this sequence:

• The first term is $a$.

• The total number of terms is $(n + 2)$.

• The last term $b$ is the $(n + 2)^{th}$ term.

Using the general term formula $a_m = ar^{m-1}$:

Calculation of Individual Means

Using the derived value of $r$, the means are calculated as follows:

Mean Position	Expression in terms of $a$ and $r$
$G_1$	$ar = a\left(\frac{b}{a}\right)^{\frac{1}{n+1}}$
$G_2$	$ar^2 = a\left(\frac{b}{a}\right)^{\frac{2}{n+1}}$
$\vdots$	$\vdots$
$G_n$	$ar^n = a\left(\frac{b}{a}\right)^{\frac{n}{n+1}}$

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The Product Property of Geometric Means

An essential property of geometric means is that the product of $n$ geometric means inserted between two numbers $a$ and $b$ is equal to the $n^{th}$ power of the single G.M. between them.

Proof:

Let the product be $P$.

$P = (ar)(ar^2)(ar^3)\dots(ar^n)$

$P = a^n \cdot r^{(1 + 2 + 3 + \dots + n)}$

Using the sum of first $n$ natural numbers $\sum n = \frac{n(n+1)}{2}$:

Substituting $r = (b/a)^{\frac{1}{n+1}}$ from equation (ii):

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Relationship Between AM and GM

The Arithmetic Mean (A.M.) and the Geometric Mean (G.M.) are two different ways to measure the central tendency of a set of numbers. A fundamental principle in algebra is that for any set of positive real numbers, the Arithmetic Mean is always greater than or equal to the Geometric Mean. This relationship is a vital tool for proving inequalities and solving optimization problems in competitive exams like JEE.

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Formal Comparison for Two Positive Numbers

Let $a$ and $b$ be any two distinct positive real numbers. We define their means as follows:

Proof of the Inequality ($A \geq G$)

To Prove: For any two positive real numbers $a$ and $b$, $\frac{a+b}{2} \geq \sqrt{ab}$.

Proof:

Let us consider the difference between the Arithmetic Mean and the Geometric Mean:

Taking the common denominator:

Since $a = (\sqrt{a})^2$ and $b = (\sqrt{b})^2$, the numerator can be rewritten as a perfect square:

We know that the square of any real number is always non-negative (greater than or equal to zero). Therefore:

This implies:

Condition for Equality

The Arithmetic Mean is exactly equal to the Geometric Mean ($A = G$) if and only if the numbers are identical.

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Product Properties of a G.P.

The relationship between terms and their geometric mean leads to useful properties regarding the product of terms in a Geometric Progression.

General Product Formula of $n$ Terms

Let a G.P. be $a, ar, ar^2, \dots, ar^{n-1}$ where $a$ is the first term and $r$ is the common ratio. Let $P_n$ denote the product of these $n$ terms.

We can group the terms of $a$ and $r$ separately:

Using the formula for the sum of the first $(n-1)$ natural numbers, $\sum(n-1) = \frac{n(n-1)}{2}$:

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Case 1: When $n$ is Odd

If the total number of terms $n$ is odd, there is exactly one middle term. The position of this term is $\left(\frac{n+1}{2}\right)^{th}$.

Derivation:

Let the middle term be $M$. Its value is given by:

Now, let us raise this middle term to the $n^{th}$ power:

Comparing this with equation (i), we see that the values are identical. Thus:

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Case 2: When $n$ is Even

If the total number of terms $n$ is even, there are two middle terms. These are located at the positions $\left(\frac{n}{2}\right)^{th}$ and $\left(\frac{n}{2} + 1\right)^{th}$.

Derivation:

Let the two middle terms be $M_1$ and $M_2$.

The Geometric Mean (G.M.) of these two middle terms is:

Now, let us raise this G.M. to the $n^{th}$ power:

This again matches the general product formula (i). Thus:

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Summation of First $n$ Natural Powers and Polynomial Series

In the study of sequences and series, we often encounter sums that do not fit the standard Arithmetic or Geometric progressions. These are known as Special Series. Specifically, the sums of the powers of the first $n$ natural numbers form the foundation for evaluating more complex series, especially when the general term ($n^{th}$ term) is a polynomial in $n$.

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1. Sum of the First $n$ Natural Numbers ($\sum n$)

The series is represented as $1 + 2 + 3 + \dots + n$. This is an Arithmetic Progression (AP) where the first term $a = 1$ and the common difference $d = 1$.

Derivation:

Using the formula for the sum of $n$ terms of an AP, $S_n = \frac{n}{2}[2a + (n-1)d]$:

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

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The Method of Differences (Principle of Telescoping)

In examinations, finding the sum of a series often requires more than just standard AP or GP formulas. We use the Method of Differences to derive summation formulas for higher powers of natural numbers. This method relies on a specific property of algebra where a sequence of terms cancels each other out, leaving only the boundary terms.

The core idea is to express the $k^{th}$ term of a series, let's call it $a_k$, as a difference of two consecutive values of some function $f(k)$. That is:

When we find the sum of this series from $k = 1$ to $n$, the expansion looks as follows:

In this summation, every positive term $f(1), f(2), \dots, f(n-1)$ is eventually cancelled by a corresponding negative term. After this systematic cancellation, the total sum simplifies dramatically to:

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Why the Identity $k(k + 1)^2 - (k - 1)k^2$ was Chosen

When our goal is to find the sum of a quadratic series (where terms involve $k^2$), we need an identity where the difference between two consecutive functional values results in a quadratic expression. A fundamental rule in calculus and finite differences is that to find the sum of a polynomial of degree $m$, we must choose a helper function $f(k)$ of degree $m+1$.

1. Determining the Function Degree

Since the target is to sum $k^2$ (which is of degree 2), our helper function $f(k)$ must be a cubic expression (degree 3). While a simple $k^3$ could work, using a "product-based" cubic makes the algebra much cleaner and easier to handle during factorization.

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2. Construction of the Helper Function $f(k)$

We construct $f(k)$ using $k$ and its immediate successor $(k+1)$. To ensure the function is cubic, we square the larger factor:

To find the previous term in the sequence, $f(k-1)$, we replace every $k$ in the function with $(k-1)$:

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3. The Difference Logic and Simplification

The beauty of this construction lies in the subtraction. When we calculate $f(k) - f(k-1)$, the $k^3$ components are designed to cancel out, leaving a lower-degree polynomial. Let us observe the logic of the Left Hand Side (LHS) without performing a full, messy expansion:

We can factor out $k^2$ from parts of the expression or simply factor out $k$ which is common to both terms:

Now, simplifying the expression inside the square brackets:

As we can see, the $k^2$ terms inside the bracket cancel out $(k^2 - k^2 = 0)$, and we are left with:

This result is perfect because it gives us exactly what we need: a term involving $k^2$ (the sum we want to find) and a term involving $k$ (the sum we already know).

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2. Sum of Squares of the First $n$ Natural Numbers ($\sum n^2$)

Let $S_n$ denote the sum of squares of the first $n$ natural numbers, which is represented as:

$S_n = 1^2 + 2^2 + 3^2 + \dots + n^2 = \sum\limits_{k=1}^{n} k^2$

Derivation of the Formula

To find the value of $S_n$, we consider the specific algebraic identity provided in the text:

By substituting the values $k = 1, 2, 3, \dots, n$ successively into identity (i), we obtain the following equations:

$\dots \dots \dots \dots \dots \dots \dots \dots \dots$

Now, we add these equations column-wise. On the left-hand side, the terms are in a telescoping pattern, where the first part of one row cancels the second part of the next row. For example, $1 \times 2^2$ in the first row is cancelled by $- (2-1) \times 2^2$ (which is $-1 \times 2^2$) in the second row.

After cancellation, only the very first part of the last row and the very last part of the first row remain:

This can be written using the summation notation as:

We know that the sum of the first $n$ natural numbers is $\sum k = \frac{n(n+1)}{2}$. Substituting this value:

Isolating the term $3 S_n$:

Taking $\frac{n(n+1)}{2}$ as a common factor from the right side:

Dividing both sides by 3, we arrive at the final formula:

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

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Answer:

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3. Sum of Cubes of the First $n$ Natural Numbers ($\sum n^3$)

The series representing the sum of the cubes of the first $n$ positive integers is given by:

$S_n = 1^3 + 2^3 + 3^3 + \dots + n^3 = \sum\limits_{k=1}^{n} k^3$

To derive the formula for this sum, we apply the Method of Differences. Since we are dealing with a power of 3, we construct a helper function of degree 4.

Derivation using the Method of Differences

We consider the specific algebraic identity involving the squares of consecutive products:

This identity is highly efficient because it isolates the $k^3$ term directly. We substitute $k = 1, 2, 3, \dots, n$ into this identity successively:

$\dots \dots \dots \dots \dots \dots \dots \dots \dots$

Adding these equations column-wise, we observe the telescoping effect on the Left Hand Side (LHS). Every positive term in one row is cancelled by a negative term in the subsequent row, such as $1^2 \cdot 2^2$ and $-1^2 \cdot 2^2$.

After all intermediate cancellations, we are left with only the first part of the last term and the last part of the first term:

Using the summation notation, this is expressed as:

Dividing both sides by 4, we isolate the sum of cubes:

We can rewrite the right-hand side as a perfect square:

$S_n = \left[ \frac{n(n + 1)}{2} \right]^2$

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Key Observation

Comparing above equation with the formula for the sum of the first $n$ natural numbers ($\sum n = \frac{n(n+1)}{2}$), we find a beautiful mathematical relationship:

In words, the sum of the cubes of the first $n$ natural numbers is equal to the square of their sum.

Answer:

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4. Summation of a Polynomial General Term

Many series that do not appear to be standard Arithmetic or Geometric Progressions have an $n^{th}$ term ($a_n$) that can be expressed as a polynomial in $n$. To find the sum of $n$ terms ($S_n$) of such a series, we use the Principle of Linearity of the Sigma ($\sum$) operator.

If the general term of a series is a polynomial of the form:

Then the sum of the first $n$ terms is given by:

By applying the property that the sum of a sum is the sum of the sums, and constants can be taken out of the sigma operator, we get:

Note that $\sum\limits_{k=1}^{n} D = \underbrace{D + D + \dots + D}_{n \text{ times}} = nD$. We then substitute the standard power formulas derived in the previous sections to obtain the final expression for $S_n$.

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Answer: