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WBB • Class XI • Mathematics • Ch 10
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Sequence and Series

Sequence and Series form the foundational backbone of discrete mathematics, mathematical analysis, and algebraic modeling. A sequence is a succession of numbers arranged in a definite order according to a specific mathematical law, formally defined as a function whose domain is the set of natural numbers. When the terms of a sequence are connected by addition signs, they form a series. Progressions are structured sequences whose terms conform to strict, regular algebraic patterns. Among these, the Arithmetic Progression (A.P.) is governed by a constant common difference, exhibiting linear growth, while the Geometric Progression (G.P.) expands through a constant common ratio, exhibiting exponential growth. The chapter provides exhaustive theoretical frameworks for calculating general terms, partial sums, and the convergence of infinite geometric series where the absolute value of the common ratio is strictly less than 1. Furthermore, it explores the harmonic progression, arithmetico-geometric series, and the profound algebraic and geometric properties of central means: Arithmetic Mean (A.M.), Geometric Mean (G.M.), and Harmonic Mean (H.M.). The universal inequality chain asserting that the Arithmetic Mean is greater than or equal to the Geometric Mean, which is in turn greater than or equal to the Harmonic Mean, serves as one of the most powerful problem-solving instruments in higher secondary algebra, optimization, and calculus.

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Why This Chapter Matters

Sequence and series are ubiquitous across modern scientific computing, financial mathematics, telecommunications, and theoretical physics. In computer science and digital signal processing, Fourier series and z-transforms decompose complex audio, visual, and sensor signals into infinite harmonic progressions. In algorithmic analysis, recurrence relations—such as the Fibonacci sequence—model divide-and-conquer runtime complexities and dynamic programming state transitions. In modern quantitative finance, compound interest, discounted cash flows, annuities, and mortgage amortizations are evaluated using the summation of finite and infinite geometric series. In theoretical physics and quantum mechanics, series expansions (such as perturbation series and Feynman diagram expansions) approximate subatomic interactions where exact analytical solutions are impossible. For students preparing for WBCHSE Class 11 board examinations and competitive examinations like WBJEE, JEE Main, and JEE Advanced, Sequence and Series is a high-frequency, heavyweight chapter that regularly features multi-concept questions bridging algebra, trigonometry, and calculus.

Chapter Roadmap & Progression

1 1. Fundamentals: Sequences, Series,...
2 2. Arithmetic Progression (A.P.): P...
3 3. Geometric Progression (G.P.): Pa...
4 4. Relationship Between A.M. and G....
5 5. Arithmetico-Geometric Progressio...
6 6. Harmonic Progression (H.P.) & Th...

Complete Concept Guide (100% Curriculum Coverage)

1. Fundamentals: Sequences, Series, Progressions & Recursive Sequences

Understanding the precise mathematical distinction between a sequence, a series, and a progression is the cornerstone of discrete algebra.

1.1 Formal Definition of a Sequence

A sequence is an ordered succession of real numbers formed according to a definite rule or mathematical law.

Formal Definition: A sequence is a function $f: \mathbb{N} \to \mathbb{R}$ whose domain is the set of natural numbers $\mathbb{N} = \{1, 2, 3, \dots\}$. The image $f(n)$ is denoted by $a_n$ or $T_n$ and is called the $n$-th term (or general term) of the sequence.
  • Finite Sequence: A sequence containing a finite number of terms, denoted $\langle a_n \rangle_{n=1}^m = \{a_1, a_2, \dots, a_m\}$.
  • Infinite Sequence: A sequence that extends indefinitely, denoted $\langle a_n \rangle_{n=1}^\infty = \{a_1, a_2, a_3, \dots\}$.
1.2 Series and Partial Sums

If $a_1, a_2, a_3, \dots, a_n, \dots$ is a sequence, then the expression obtained by connecting the terms with addition signs is called a series:

$$S = a_1 + a_2 + a_3 + \dots + a_n + \dots = \sum_{k=1}^\infty a_k$$

The sum of the first $n$ terms is known as the $n$-th partial sum:

$$S_n = a_1 + a_2 + \dots + a_n = \sum_{k=1}^n a_k$$
1.3 Recursive Sequences & The Fibonacci Sequence

A sequence is recursive (or defined by a recurrence relation) when each term after the initial terms is defined as a function of the preceding terms.

Classic Example (The Fibonacci Sequence): $$a_1 = 1, \quad a_2 = 1, \quad a_n = a_{n-1} + a_{n-2} \quad \text{for all } n \ge 3$$ The terms generated are: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \dots$

2. Arithmetic Progression (A.P.): Properties, Sums & Arithmetic Means

An Arithmetic Progression is a sequence in which the difference between any term and its immediate predecessor is a constant quantity.

2.1 Definition, General Term & Common Difference

A sequence $a_1, a_2, a_3, \dots$ is an Arithmetic Progression (A.P.) if and only if:

$$a_{n+1} - a_n = d = \text{constant} \quad (\text{called the Common Difference})$$
General Term ($n$-th term): If $a$ is the first term and $d$ is the common difference, then: $$\mathbf{T_n = a + (n - 1)d}$$ The $p$-th term from the end of a finite A.P. having $m$ terms is $T_{m - p + 1} = l - (p - 1)d$, where $l$ is the last term.
2.2 Sum of First $n$ Terms ($S_n$)

The sum of the first $n$ terms of an A.P. is derived by Gauss\'s pairing method:

$$S_n = a + (a + d) + (a + 2d) + \dots + [a + (n - 1)d]$$

Writing in reverse order and adding corresponding terms:

$$2 S_n = [2a + (n - 1)d] + [2a + (n - 1)d] + \dots \text{ ($n$ times)}$$
$$\mathbf{S_n = \frac{n}{2} [2a + (n - 1)d] = \frac{n}{2} [a + l]}$$ where $l = a + (n - 1)d$ is the last term.
2.3 Key Operational Rules in A.P.
Scenario Standard Choice of Terms Common Difference Primary Advantage
3 terms in A.P. $a - d, \; a, \; a + d$ $d$ Sum $= 3a$ (immediately yields $a$)
4 terms in A.P. $a - 3d, \; a - d, \; a + d, \; a + 3d$ $2d$ Sum $= 4a$ (odd powers of $d$ cancel)
5 terms in A.P. $a - 2d, \; a - d, \; a, \; a + d, \; a + 2d$ $d$ Sum $= 5a$
2.4 Arithmetic Mean (A.M.) and Insertion of Means
  • Single A.M.: If $a, A, b$ are in A.P., then $A - a = b - A \implies \mathbf{A = \frac{a + b}{2}}$.
  • Insertion of $n$ Arithmetic Means ($A_1, A_2, \dots, A_n$) between $a$ and $b$:

    The resulting sequence $a, A_1, A_2, \dots, A_n, b$ forms an A.P. with $(n + 2)$ terms. The last term $b = a + (n + 2 - 1)d = a + (n + 1)d$.

    $$\mathbf{d = \frac{b - a}{n + 1}}, \quad A_k = a + k d = a + k \left( \frac{b - a}{n + 1} \right)$$ Sum of $n$ inserted A.M.s: $$\sum_{k=1}^n A_k = \frac{n}{2}(A_1 + A_n) = \frac{n}{2}(a + b) = \mathbf{n \cdot A}$$

3. Geometric Progression (G.P.): Partial Sums, Infinite Series & Means

A Geometric Progression is a sequence in which each term after the first is obtained by multiplying the previous term by a fixed non-zero constant.

3.1 Definition and General Term

A sequence $a_1, a_2, a_3, \dots$ is a Geometric Progression (G.P.) if and only if:

$$\frac{a_{n+1}}{a_n} = r = \text{constant} \quad (\text{called the Common Ratio}, r \neq 0)$$
General Term ($n$-th term): If $a$ is the first term and $r$ is the common ratio: $$\mathbf{T_n = a \cdot r^{n-1}}$$
3.2 Sum of First $n$ Terms ($S_n$)
$$S_n = a + ar + ar^2 + \dots + ar^{n-1}$$ $$r S_n = ar + ar^2 + \dots + ar^n$$

Subtracting the second equation from the first:

$$S_n (1 - r) = a(1 - r^n) \implies \mathbf{S_n = \frac{a(1 - r^n)}{1 - r} = \frac{a(r^n - 1)}{r - 1} \quad (r \neq 1)}$$

When $r = 1$, $S_n = a + a + \dots + a = \mathbf{n a}$.

3.3 Sum of an Infinite G.P. ($S_\infty$) and Convergence

If the common ratio satisfies $|r| < 1$, then as $n \to \infty$, $r^n \to 0$:

$$\mathbf{S_\infty = \lim_{n \to \infty} S_n = \frac{a}{1 - r}, \quad \text{strictly valid for } |r| < 1}$$ Divergence Note: If $|r| \ge 1$, the infinite series does not converge to a finite sum.

Application to Recurring Decimals:

$$0.\dot{3}\dot{7} = 0.373737\dots = \frac{37}{100} + \frac{37}{100^2} + \frac{37}{100^3} + \dots$$

Here $a = \frac{37}{100}$ and $r = \frac{1}{100} < 1$:

$$S_\infty = \frac{37/100}{1 - 1/100} = \frac{37/100}{99/100} = \mathbf{\frac{37}{99}}$$
3.4 Geometric Mean (G.M.) and Insertion of Means
  • Single G.M.: If $a, G, b$ are positive numbers in G.P., then $\frac{G}{a} = \frac{b}{G} \implies \mathbf{G = \sqrt{ab}}$.
  • Insertion of $n$ Geometric Means ($G_1, G_2, \dots, G_n$) between $a$ and $b$:

    The sequence $a, G_1, G_2, \dots, G_n, b$ forms a G.P. of $(n + 2)$ terms, with $b = a r^{n+1}$:

    $$\mathbf{r = \left( \frac{b}{a} \right)^{\frac{1}{n+1}}}, \quad G_k = a r^k = a \left( \frac{b}{a} \right)^{\frac{k}{n+1}}$$ Product of $n$ inserted G.M.s: $$\prod_{k=1}^n G_k = G_1 \cdot G_2 \dots G_n = (\sqrt{ab})^n = \mathbf{G^n}$$

4. Relationship Between A.M. and G.M.: Proofs, Quadratic Roots & Inequalities

The inequality $A \ge G$ is one of the most powerful and ubiquitous tools in algebraic optimization and inequalities.

4.1 The Fundamental AM-GM Inequality
Theorem: For any two positive real numbers $a$ and $b$, their Arithmetic Mean $A = \frac{a + b}{2}$ and Geometric Mean $G = \sqrt{ab}$ satisfy: $$\mathbf{A \ge G}$$ with equality $A = G$ holding if and only if $a = b$.

Algebraic Proof:

$$A - G = \frac{a + b}{2} - \sqrt{ab} = \frac{a + b - 2\sqrt{ab}}{2} = \frac{(\sqrt{a} - \sqrt{b})^2}{2}$$

Since the square of any real number is non-negative, $(\sqrt{a} - \sqrt{b})^2 \ge 0$, which proves:

$$A - G \ge 0 \implies \mathbf{A \ge G}$$

Equality holds if and only if $(\sqrt{a} - \sqrt{b})^2 = 0 \iff \sqrt{a} = \sqrt{b} \iff a = b$.

4.2 Semicircle Geometric Proof of AM ≥ GM

Construct a semicircle whose diameter is $AB = a + b$. The radius of this circle is:

$$R = \frac{a + b}{2} = A \quad (\text{Arithmetic Mean})$$

Erect a vertical perpendicular from the division point $D$ (where $AD = a$ and $DB = b$) to meet the semicircle at point $P$. By the geometric mean theorem for right triangles, the altitude is:

$$DP = \sqrt{AD \cdot DB} = \sqrt{ab} = G \quad (\text{Geometric Mean})$$

Since the vertical perpendicular $DP$ cannot exceed the circle\'s radius $R$ (altitude $\le$ hypotenuse/radius), it follows geometrically that $\mathbf{A \ge G}$, with equality occurring when $D$ coincides with the circle\'s center ($a = b$).

4.3 Formulating Quadratic Equations with Roots $a$ and $b$

Given the Arithmetic Mean $A$ and Geometric Mean $G$ of two positive numbers $a$ and $b$:

$$\text{Sum of roots: } a + b = 2A, \quad \text{Product of roots: } a b = G^2$$

The quadratic equation having roots $a$ and $b$ is:

$$\mathbf{x^2 - 2Ax + G^2 = 0}$$

Solving via the quadratic formula:

$$\mathbf{a, b = A \pm \sqrt{A^2 - G^2}}$$

5. Arithmetico-Geometric Progression (A.G.P.) & Power Sums of Natural Numbers

Many physical and financial phenomena combine additive and multiplicative growth simultaneously, giving rise to Arithmetico-Geometric Series.

5.1 Arithmetico-Geometric Progression (A.G.P.)

An A.G.P. is a sequence whose $n$-th term is the product of the corresponding terms of an A.P. and a G.P.:

$$a, \; (a + d)r, \; (a + 2d)r^2, \; \dots, \; [a + (n - 1)d]r^{n-1}$$

To find the sum of first $n$ terms $S_n$:

$$S_n = a + (a + d)r + (a + 2d)r^2 + \dots + [a + (n - 1)d]r^{n-1}$$ $$r S_n = ar + (a + d)r^2 + \dots + [a + (n - 2)d]r^{n-1} + [a + (n - 1)d]r^n$$

Subtracting yields a standard G.P. of $n - 1$ terms:

$$(1 - r) S_n = a + \left[ d r + d r^2 + \dots + d r^{n-1} \right] - [a + (n - 1)d]r^n$$
5.2 Sum of an Infinite A.G.P. ($|r| < 1$)

When $|r| < 1$, as $n \to \infty$, $r^n \to 0$ and $n r^n \to 0$:

$$\mathbf{S_\infty = \frac{a}{1 - r} + \frac{d r}{(1 - r)^2}, \quad (|r| < 1)}$$
5.3 Sum of Powers of First $n$ Natural Numbers
Formula Name Series Expansion Closed-Form Result
First $n$ Integers $\sum_{k=1}^n k = 1 + 2 + 3 + \dots + n$ $$\mathbf{\frac{n(n + 1)}{2}}$$
Sum of Squares $\sum_{k=1}^n k^2 = 1^2 + 2^2 + 3^2 + \dots + n^2$ $$\mathbf{\frac{n(n + 1)(2n + 1)}{6}}$$
Sum of Cubes $\sum_{k=1}^n k^3 = 1^3 + 2^3 + 3^3 + \dots + n^3$ $$\mathbf{\left[\frac{n(n + 1)}{2}\right]^2 = \left(\sum_{k=1}^n k\right)^2}$$

6. Harmonic Progression (H.P.) & The Unified Means Hierarchy ($A \ge G \ge H$)

The harmonic progression completes the classical trilogy of progressions and establishes the complete hierarchy of central means.

6.1 Harmonic Progression (H.P.)

A sequence $a_1, a_2, a_3, \dots$ of non-zero real numbers is in Harmonic Progression (H.P.) if and only if their reciprocals form an Arithmetic Progression (A.P.):

$$\frac{1}{a_1}, \; \frac{1}{a_2}, \; \frac{1}{a_3}, \; \dots \quad \text{are in A.P.}$$

If the corresponding A.P. has first term $1/a$ and common difference $d$, the $n$-th term of the H.P. is:

$$\mathbf{T_n = \frac{1}{\frac{1}{a} + (n - 1)d}}$$

Crucial Warning: There is no general algebraic closed-form formula for the sum of $n$ terms of an H.P. To solve problems involving H.P., always invert the terms into an A.P.

6.2 Harmonic Mean (H.M.)

If $a, H, b$ are in H.P., then $\frac{1}{a}, \frac{1}{H}, \frac{1}{b}$ are in A.P.:

$$\frac{2}{H} = \frac{1}{a} + \frac{1}{b} = \frac{a + b}{ab} \implies \mathbf{H = \frac{2ab}{a + b}}$$
6.3 The Geometric Bridge: $G^2 = A \cdot H$

For any two positive numbers $a$ and $b$ with A.M. $A$, G.M. $G$, and H.M. $H$:

$$A \cdot H = \left(\frac{a + b}{2}\right) \left(\frac{2ab}{a + b}\right) = ab = (\sqrt{ab})^2 = G^2$$
$$\mathbf{G^2 = A \cdot H \iff G = \sqrt{A \cdot H}}$$ This proves that the Geometric Mean $G$ is the geometric mean of the Arithmetic Mean $A$ and Harmonic Mean $H$! Thus, $A, G, H$ form a G.P.
6.4 The Complete Means Hierarchy: $A \ge G \ge H$

Since $A \ge G$ and $G^2 = A H \implies \frac{G}{H} = \frac{A}{G} \ge 1 \implies G \ge H$:

$$\mathbf{A \ge G \ge H}$$ Equality $A = G = H$ holds if and only if $a = b$.

Key Formulas, Identities & Theorems

Arithmetic Progression Sum
(n/2)[2a + (n - 1)d]
l is the last term. General term relation: T_n = S_n - S_{n-1}.
Geometric Progression Partial Sum
a(1 - rⁿ) / (1 - r)
For r = 1, S_n = na.
Sum of Infinite G.P.
a / (1 - r)
Strictly valid only when the absolute value of r is less than 1. Diverges if |r| ≥ 1.
AM-GM Fundamental Inequality
(a + b)/2 ≥ √(ab)
Equality holds if and only if a = b. Roots formula: a, b = A ± √(A² - G²).
Infinite AGP Sum
a/(1 - r) + dr/(1 - r)²
Derived by subtracting r·S_∞ from S_∞.
Unified Means Hierarchy
A ≥ G ≥ H and G² = AH
A, G, H form a G.P. Harmonic Mean H = 2ab / (a + b).

Conceptual Solved Examples & Case Studies

Example 1
If the sum of \(n\) terms of an A.P. is \(S_n = 3n^2 + 5n\), find its \(n\)-th term and the common difference. [2 marks]
Step-by-Step Solution:
Solution: Given: $S_n = 3n^2 + 5n$. For $n \ge 2$, the $n$-th term is given by: $$T_n = S_n - S_{n-1}$$ Compute $S_{n-1}$: $$S_{n-1} = 3(n - 1)^2 + 5(n - 1) = 3(n^2 - 2n + 1) + 5n - 5 = 3n^2 - 6n + 3 + 5n - 5 = 3n^2 - n - 2$$ Now subtract: $$T_n = (3n^2 + 5n) - (3n^2 - n - 2) = 3n^2 + 5n - 3n^2 + n + 2 = \mathbf{6n + 2}$$ Verification for $n = 1$: $$T_1 = 6(1) + 2 = 8, \quad S_1 = 3(1)^2 + 5(1) = 8 \quad (\text{Matches!})$$ Common Difference $d$: $$d = T_n - T_{n-1} = [6n + 2] - [6(n - 1) + 2] = 6n + 2 - (6n - 4) = \mathbf{6}$$ Hence, the $n$-th term is $6n + 2$ and the common difference is $6$.
Example 2
The sum of three numbers in A.P. is \(21\) and their product is \(315\). Find the numbers. [3 marks]
Step-by-Step Solution:
Solution: Let the three numbers in A.P. be: $$a - d, \quad a, \quad a + d$$
Condition 1: Sum is $21$: $$(a - d) + a + (a + d) = 21 \implies 3a = 21 \implies \mathbf{a = 7}$$
Condition 2: Product is $315$: $$(a - d) \cdot a \cdot (a + d) = 315$$ Substitute $a = 7$: $$(7 - d) \cdot 7 \cdot (7 + d) = 315$$ Divide both sides by $7$: $$(7 - d)(7 + d) = \frac{315}{7} = 45$$ $$49 - d^2 = 45 \implies d^2 = 49 - 45 = 4 \implies \mathbf{d = \pm 2}$$
Forming the Numbers:
  • If $d = 2$: Numbers are $7 - 2, 7, 7 + 2 \implies \mathbf{5, 7, 9}$.
  • If $d = -2$: Numbers are $7 - (-2), 7, 7 + (-2) \implies \mathbf{9, 7, 5}$.
Hence, the three numbers are $5, 7, 9$.
Example 3
Find the sum of the infinite geometric series: \(5 + \frac{10}{3} + \frac{20}{9} + \frac{40}{27} + \dots\) [3 marks]
Step-by-Step Solution:
Solution: Given series: $5 + \frac{10}{3} + \frac{20}{9} + \frac{40}{27} + \dots$ Here, the first term is $a = 5$. The common ratio is: $$r = \frac{T_2}{T_1} = \frac{10/3}{5} = \frac{10}{15} = \frac{2}{3}$$ Check convergence condition: $$|r| = \left|\frac{2}{3}\right| = \frac{2}{3} < 1$$ Since the absolute value of the common ratio is strictly less than $1$, the infinite series converges. Applying the infinite G.P. sum formula: $$S_\infty = \frac{a}{1 - r} = \frac{5}{1 - \frac{2}{3}} = \frac{5}{\frac{1}{3}} = 5 \times 3 = \mathbf{15}$$ Hence, the sum of the infinite series is $15$.
Example 4
Between \(1\) and \(31\), \(m\) arithmetic means have been inserted in such a way that the ratio of the \(7\)-th mean to the \((m - 1)\)-th mean is \(5 : 9\). Find the value of \(m\). [4 marks]
Step-by-Step Solution:
Solution: Let the $m$ arithmetic means inserted between $1$ and $31$ be $A_1, A_2, \dots, A_m$. The complete sequence: $$1, A_1, A_2, \dots, A_m, 31$$ forms an A.P. with total terms = $m + 2$. Here $a = 1$ and last term $T_{m+2} = 31$: $$31 = 1 + (m + 2 - 1)d \implies 30 = (m + 1)d \implies \mathbf{d = \frac{30}{m + 1}}$$ The $k$-th arithmetic mean is: $$A_k = a + k d = 1 + k \left( \frac{30}{m + 1} \right) = \frac{m + 1 + 30k}{m + 1}$$ Therefore: $$A_7 = \frac{m + 1 + 30(7)}{m + 1} = \frac{m + 211}{m + 1}$$ $$A_{m-1} = \frac{m + 1 + 30(m - 1)}{m + 1} = \frac{m + 1 + 30m - 30}{m + 1} = \frac{31m - 29}{m + 1}$$ Given the ratio: $$\frac{A_7}{A_{m-1}} = \frac{m + 211}{31m - 29} = \frac{5}{9}$$ Cross-multiplying: $$9(m + 211) = 5(31m - 29)$$ $$9m + 1899 = 155m - 145$$ $$1899 + 145 = 155m - 9m$$ $$2044 = 146m \implies \mathbf{m = \frac{2044}{146} = 14}$$ Hence, the value of $m$ is $14$.
Example 5
If \(A\) and \(G\) be the Arithmetic Mean and Geometric Mean between two positive numbers, prove that the numbers are \(A \pm \sqrt{(A + G)(A - G)}\). [4 marks]
Step-by-Step Solution:
Solution: Let the two positive numbers be $a$ and $b$. By definition of Arithmetic Mean and Geometric Mean: $$A = \frac{a + b}{2} \implies a + b = 2A \quad \text{--- (Equation 1)}$$ $$G = \sqrt{ab} \implies ab = G^2 \quad \text{--- (Equation 2)}$$ We know the algebraic identity: $$(a - b)^2 = (a + b)^2 - 4ab$$ Substitute Equation 1 and Equation 2: $$(a - b)^2 = (2A)^2 - 4G^2 = 4A^2 - 4G^2 = 4(A^2 - G^2)$$ Taking the square root on both sides: $$a - b = \pm 2\sqrt{A^2 - G^2} \quad \text{--- (Equation 3)}$$ Factoring the difference of squares under the radical: $$a - b = \pm 2\sqrt{(A + G)(A - G)}$$ Adding Equation 1 and Equation 3: $$2a = 2A \pm 2\sqrt{(A + G)(A - G)} \implies a = A \pm \sqrt{(A + G)(A - G)}$$ Subtracting Equation 3 from Equation 1: $$2b = 2A \mp 2\sqrt{(A + G)(A - G)} \implies b = A \mp \sqrt{(A + G)(A - G)}$$ Hence, the two numbers are: $$\mathbf{A \pm \sqrt{(A + G)(A - G)}}$$ Hence proved.
Example 6
(i) Find the sum to \(n\) terms of the series: \(1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \dots + n(n + 1)\). [2.5 marks] (ii) Find the sum of the infinite Arithmetico-Geometric series: \(1 + \frac{4}{5} + \frac{7}{5^2} + \frac{10}{5^3} + \dots\) [2.5 marks] [5 marks total]
Step-by-Step Solution:
Solution:
Part (i): Sum to $n$ terms of $\sum k(k + 1)$: [2.5 marks] The $k$-th term of the series is: $$T_k = k(k + 1) = k^2 + k$$ Therefore, the sum to $n$ terms is: $$S_n = \sum_{k=1}^n T_k = \sum_{k=1}^n (k^2 + k) = \sum_{k=1}^n k^2 + \sum_{k=1}^n k$$ Using the standard power sum formulas: $$S_n = \frac{n(n + 1)(2n + 1)}{6} + \frac{n(n + 1)}{2}$$ Factor out the common term $\frac{n(n + 1)}{2}$: $$S_n = \frac{n(n + 1)}{2} \left[ \frac{2n + 1}{3} + 1 \right] = \frac{n(n + 1)}{2} \left[ \frac{2n + 1 + 3}{3} \right] = \frac{n(n + 1)(2n + 4)}{6}$$ $$S_n = \frac{2n(n + 1)(n + 2)}{6} = \mathbf{\frac{n(n + 1)(n + 2)}{3}}$$
Part (ii): Sum of the infinite A.G.P. $S = 1 + \frac{4}{5} + \frac{7}{5^2} + \frac{10}{5^3} + \dots$: [2.5 marks] Let $S = 1 + \frac{4}{5} + \frac{7}{5^2} + \frac{10}{5^3} + \dots \quad \text{--- (Equation 1)}$ Here, the numerators $1, 4, 7, 10, \dots$ form an A.P. with $a = 1, d = 3$. The denominators form a G.P. with common ratio $r = \frac{1}{5}$. Multiply Equation 1 by $r = \frac{1}{5}$: $$\frac{1}{5} S = \frac{1}{5} + \frac{4}{5^2} + \frac{7}{5^3} + \dots \quad \text{--- (Equation 2)}$$ Subtract Equation 2 from Equation 1 (aligning identical denominators): $$S - \frac{1}{5} S = 1 + \left( \frac{4}{5} - \frac{1}{5} \right) + \left( \frac{7}{5^2} - \frac{4}{5^2} \right) + \left( \frac{10}{5^3} - \frac{7}{5^3} \right) + \dots$$ $$\frac{4}{5} S = 1 + \frac{3}{5} + \frac{3}{5^2} + \frac{3}{5^3} + \dots$$ $$\frac{4}{5} S = 1 + 3 \left[ \frac{1}{5} + \frac{1}{5^2} + \frac{1}{5^3} + \dots \right]$$ The bracketed term is an infinite G.P. with first term $\frac{1}{5}$ and common ratio $\frac{1}{5} < 1$: $$\text{Sum} = \frac{1/5}{1 - 1/5} = \frac{1/5}{4/5} = \frac{1}{4}$$ Therefore: $$\frac{4}{5} S = 1 + 3 \left( \frac{1}{4} \right) = 1 + \frac{3}{4} = \frac{7}{4}$$ Solve for $S$: $$S = \frac{7}{4} \times \frac{5}{4} = \mathbf{\frac{35}{16}}$$ Hence, the sum of the infinite series is $\frac{35}{16}$.

Common Misconceptions & Examiner Traps

Common Misconception

Applying the Infinite G.P. Formula when |r| ≥ 1

Scientific Reality & Correction

The infinite sum formula S_∞ = a / (1 - r) is valid ONLY when |r| < 1.

Common Misconception

Confusing Term Number with Number of Inserted Means

Scientific Reality & Correction

Since b is the (n + 2)-th term, b = a + (n + 1)d, so d = (b - a)/(n + 1).

Common Misconception

Calculating T_n from S_n for n = 1 Using T_n = S_n - S_{n-1}

Scientific Reality & Correction

The recurrence T_n = S_n - S_{n-1} holds strictly for n ≥ 2. For n = 1, T₁ is simply S₁.

Common Misconception

Incorrect Alignment in Arithmetico-Geometric Summation

Scientific Reality & Correction

When multiplying S by r, shift the series rightward so that terms with like powers of r align.

Common Misconception

Applying AM ≥ GM to Negative Real Numbers

Scientific Reality & Correction

AM ≥ GM is strictly established only for POSITIVE real numbers a, b > 0.

Sequence & Series Architecture, A.P., G.P. & Means Hierarchy Diagram

WBCHSE CLASS 11 • ALGEBRA • CHAPTER 10 Sequence & Series Architecture: A.P., G.P. & Means Hierarchy Arithmetic & Geometric Progressions • AM-GM-HM Inequality • Special Series Summations ★ A ≥ G ≥ H | G² = A · H AM-GM-HM Unified Chain ARITHMETIC PROGRESSION (A.P.) Arithmetic Progression (A.P.) Tₙ = a + (n - 1)d Sₙ = (n/2)[2a + (n - 1)d] = (n/2)(a + l) l = last term | Tₙ = Sₙ - Sₙ₋₁ A.M. A = (a + b)/2 | Σ Aₖ = n · A GEOMETRIC PROGRESSION (G.P.) Tₙ = a · rⁿ⁻¹ Sₙ = a(1 - rⁿ)/(1 - r) (r ≠ 1) S_∞ = a / (1 - r) (|r| < 1) Valid if and only if |r| < 1 (diverges if |r| ≥ 1) G.M. G = √(ab) | Π Gₖ = Gⁿ Recurring Decimals: 0.3̄7̄ = 37/99 via infinite G.P. AM-GM GEOMETRIC PROOF DIAGRAM Semicircle with diameter a + b: Radius A ≥ Altitude G Center C a b G = √(ab) A = (a+b)/2 UNIFIED MEANS HIERARCHY A ≥ G ≥ H (Equality iff a = b) G² = A · H ⇒ A, G, H form a G.P. Equality holds if and only if a = b Quadratic Equation with Roots a, b: x² - 2Ax + G² = 0 Roots a, b = A ± √(A² - G²) POWER SUMS OF NATURAL NUMBERS Standard Power Sum Formulas Σ k = n(n + 1)/2 Σ k² = n(n + 1)(2n + 1)/6 Σ k³ = [n(n + 1)/2]² = (Σ k)² ARITHMETICO-GEOMETRIC PROGRESSION (A.G.P.) Form: a, (a+d)r, (a+2d)r², (a+3d)r³, ... S_∞ = a/(1 - r) + dr/(1 - r)² (|r| < 1) Obtained by subtracting r·S_∞ from S_∞ HARMONIC PROGRESSION (H.P.) Reciprocals form an Arithmetic Progression H = 2ab / (a + b) | G² = A · H

Chapter Summary & 10 Key Takeaways

Takeaway 1
A sequence is a function f: ℕ → ℝ; connecting sequence terms with addition signs produces a series.
Takeaway 2
Arithmetic Progression (A.P.): Tₙ = a + (n - 1)d; sum Sₙ = (n/2)[2a + (n - 1)d] = (n/2)(a + l).
Takeaway 3
For any series, the n-th term is Tₙ = Sₙ - Sₙ₋₁ for n ≥ 2, with T₁ = S₁.
Takeaway 4
Single A.M. is A = (a + b)/2; inserting n A.M.s yields common difference d = (b - a)/(n + 1).
Takeaway 5
Geometric Progression (G.P.): Tₙ = a · rⁿ⁻¹; sum Sₙ = a(1 - rⁿ)/(1 - r) for r ≠ 1.
Takeaway 6
An infinite G.P. converges to S_∞ = a / (1 - r) if and only if |r| < 1.
Takeaway 7
Single G.M. is G = √(ab); product of n inserted G.M.s is Gⁿ = (ab)^{n/2}.
Takeaway 8
For positive real numbers, Arithmetic Mean is greater than or equal to Geometric Mean (A ≥ G), with equality iff a = b.
Takeaway 9
Harmonic Progression (H.P.) consists of numbers whose reciprocals form an A.P.; H.M. H = 2ab / (a + b).
Takeaway 10
Unified Means Hierarchy: A ≥ G ≥ H and G² = A · H (A, G, H form a G.P.).

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
What is the sum of the first 20 odd natural numbers 1 + 3 + 5 + ... + 39?
Reveal Answer & Explanation
Answer: The sum is n² = 20² = 400.
Use the sum formula for an A.P. or recall that the sum of the first n odd numbers is n².
2
What is the sum of the infinite series 1 + 1/2 + 1/4 + 1/8 + ...?
Reveal Answer & Explanation
Answer: S_∞ = 1 / (1 - 1/2) = 1 / (1/2) = 2.
Use S_∞ = a / (1 - r) with a = 1 and r = 1/2.
3
If the A.M. between two numbers is 10 and their G.M. is 8, find the numbers.
Reveal Answer & Explanation
Answer: a, b = 10 ± √(100 - 64) = 10 ± √36 = 10 ± 6. The numbers are 16 and 4.
Use the formula a, b = A ± √(A² - G²).
4
What is the value of 1² + 2² + 3² + ... + 10²?
Reveal Answer & Explanation
Answer: 10 × 11 × 21 / 6 = 2310 / 6 = 385.
Use n(n + 1)(2n + 1)/6 with n = 10.
5
If A, G, H are the means between two numbers, what is the relation between them?
Reveal Answer & Explanation
Answer: G² = A · H (they form a G.P.) and A ≥ G ≥ H.
Recall the geometric relationship connecting G to A and H.
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