Chapter 8 of 14

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CBSE Class 11 Mathematics★ 6-8 marks · CBSE Class 11 Mathematics Chapter 8

Sequences and Series

2, 4, 6, 8... (AP). 3, 9, 27, 81... (GP). This chapter covers Arithmetic Progressions and Geometric Progressions — how to find any term, sum the series, and insert means between terms. CBSE Class 11 Mathematics Chapter 8.

⏱ 26 min readMedium difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Find the nth term and sum of the first n terms of an Arithmetic Progression
2Find the nth term and sum (finite and infinite) of a Geometric Progression
3Insert Arithmetic Means and Geometric Means between two given numbers
4Apply the AM ≥ GM inequality for positive numbers
5Evaluate sums of special series: Σn, Σn², Σn³

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Why this chapter matters

Arithmetic and Geometric Progressions appear in nearly every board and JEE paper — from finding the nth term to summing series. The special series formulas (Σn, Σn², Σn³) are used extensively in calculus and in advanced counting problems.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Relations and Functions

A sequence is a function from N to R; function notation and algebraic manipulation required

Sequences and Series

"A sequence is a list. A series is a sum. One tells a story. The other gives the total."

1. Chapter Overview

A SEQUENCE is an ordered list of numbers following a RULE. A SERIES is the SUM of the terms of a sequence. This chapter covers: ARITHMETIC PROGRESSIONS (AP — common difference), GEOMETRIC PROGRESSIONS (GP — common ratio), their sums, arithmetic and geometric means, and the sum of SPECIAL SERIES (Σn, Σn², Σn³).

2. Sequences — Basic Concepts

A sequence: a₁, a₂, a₃, ..., aₙ
A series: a₁ + a₂ + a₃ + ... + aₙ
Finite sequence: has a LAST term
Infinite sequence: goes on FOREVER

3. Arithmetic Progression (AP)

Definition

A sequence where the DIFFERENCE between consecutive terms is CONSTANT
This constant difference d = COMMON DIFFERENCE
d = a₂ — a₁ = a₃ — a₂ = ...

nth Term (General Term)

$a_{n} = a_{1} + (n - 1) d$ where a₁ = first term, d = common difference

Sum of First n Terms (Sₙ)

$S_{n} = \frac{n}{2} [2 a_{1} + (n - 1) d]$ OR: $S_{n} = \frac{n}{2} (a_{1} + a_{n})$ (First term + Last term) × n/2

Arithmetic Mean (AM) Between Two Numbers

AM of a and b = (a + b)/2
To insert n Arithmetic Means between a and b: the n AMs combined with a and b form an AP of (n+2) terms. Find d. Insert means.

4. Geometric Progression (GP)

Definition

A sequence where the RATIO of consecutive terms is CONSTANT
This constant ratio r = COMMON RATIO
r = a₂/a₁ = a₃/a₂ = ...

nth Term (General Term)

$a_{n} = a_{1} \times r^{n - 1}$

Sum of First n Terms (Sₙ)

$S_{n} = \frac{a _{1} ( r ^{n} - 1 )}{r - 1} for r > 1$ $S_{n} = \frac{a _{1} ( 1 - r ^{n} )}{1 - r} for r < 1$

Sum of an INFINITE GP (only when |r| < 1)

$S_{\infty} = \frac{a}{1 - r}$ The terms get smaller and smaller → their sum approaches a FINITE limit.

Geometric Mean (GM) Between Two Numbers

GM of a and b = √(ab) (for positive a, b)
To insert n Geometric Means between a and b: the n GMs + a and b form a GP of (n+2) terms.

5. Relationship Between AM and GM

For any two positive numbers: AM ≥ GM
Equality holds ONLY when a = b

6. Sum of Special Series

Σn = 1 + 2 + 3 + ... + n = n(n+1)/2
Σn² = 1² + 2² + 3² + ... + n² = n(n+1)(2n+1)/6
Σn³ = 1³ + 2³ + 3³ + ... + n³ = [n(n+1)/2]² = (Σn)²

7. Exam Focus

AP — nth term formula, sum formula (Sₙ)
GP — nth term formula, sum formula (Sₙ), infinite GP sum (|r|<1)
Inserting Arithmetic Means and Geometric Means
AM ≥ GM (for positive numbers)
Sum of Σn, Σn², Σn³

8. Key Formulas Summary

Concept	Formula
AP nth term	aₙ = a₁ + (n-1)d
AP sum	Sₙ = n/2[2a₁ + (n-1)d] = n/2(a₁ + aₙ)
GP nth term	aₙ = a₁ rⁿ⁻¹
GP sum (finite)	Sₙ = a₁(rⁿ-1)/(r-1) (r>1)
GP sum (infinite)	S∞ = a/(1-r) (
Σn	n(n+1)/2
Σn²	n(n+1)(2n+1)/6
Σn³	[n(n+1)/2]²

9. Conclusion

Sequences and series describe PATTERNS:

AP: Constant addition. 3, 7, 11, 15... (d=4). Sum is the average of first and last, times n.
GP: Constant multiplication. 2, 6, 18, 54... (r=3). Exponential growth (or decay, if |r|<1).
INFINITE GP: When |r|<1, adding forever gives a FINITE number. A beautiful limit.
SPECIAL SUMS: Σn, Σn², Σn³ — formulas you'll use in calculus.

'A sequence is a function whose domain is the natural numbers.' Sequences are the discrete-building-blocks that continuous functions smooth over.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

AP: nth Term

aₙ = a + (n−1)d

a = first term, d = common difference = a₂−a₁

AP: Sum of n Terms

Sₙ = n/2·[2a + (n−1)d] = n/2·(a + aₙ)

Second form used when first and last terms are known; both are equivalent

GP: nth Term

aₙ = a·rⁿ⁻¹

a = first term, r = common ratio = a₂/a₁

GP: Sum of n Terms

Sₙ = a(rⁿ−1)/(r−1) for r ≠ 1; Sₙ = na for r = 1

Use a(1−rⁿ)/(1−r) when |r|<1 to avoid negative denominators

GP: Sum of Infinite Terms

S∞ = a/(1−r), valid only when |r| < 1

If |r| ≥ 1 the series diverges and has no finite sum

Arithmetic Mean

AM of a and b = (a+b)/2

If n AMs are inserted between a and b, the common difference d = (b−a)/(n+1)

Geometric Mean

GM of a and b = √(ab) for positive a, b

If n GMs are inserted between a and b, the common ratio r = (b/a)^(1/(n+1))

Special Series

Σk = n(n+1)/2; Σk² = n(n+1)(2n+1)/6; Σk³ = [n(n+1)/2]²

Note: Σk³ = (Σk)² — the sum of cubes equals the square of the sum of natural numbers

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Using the infinite GP sum S∞ = a/(1−r) when |r| ≥ 1

✓ S∞ = a/(1−r) is only valid when |r| < 1 (terms are shrinking). When |r| ≥ 1 the terms grow or stay constant and the series has no finite sum.

WATCH OUT

✗ Confusing the AP sum formula: using n terms but plugging in n−1

✓ Sₙ = n/2·[2a+(n−1)d]. The (n−1) inside the bracket is the multiplier of d, not the number of terms. The number of terms is n.

WATCH OUT

✗ Treating the common difference as a₁/a₂ instead of a₂−a₁

✓ Common difference d = aₙ₊₁ − aₙ (subtract consecutive terms). Common ratio r = aₙ₊₁/aₙ (divide). d is a difference; r is a ratio.

WATCH OUT

✗ Forgetting to verify a sequence is AP/GP before applying formulas

✓ Check: constant difference between consecutive terms → AP. Constant ratio → GP. If neither applies, the sequence is general and these formulas are invalid.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Ex 8.1

Exercise 8.1

AP — nth term, sum, inserting arithmetic means, AM problems

Questions

Ex 8.2

Exercise 8.2

GP — nth term, sum of finite and infinite series, inserting geometric means

Questions

Ex 8.3

Exercise 8.3

Special series — Σn, Σn², Σn³ — combined with AP/GP

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· AP

Find the 20th term and sum of first 20 terms of the AP: 3, 7, 11, 15, ...

▶ Show solution

a = 3, d = 7−3 = 4. a₂₀ = 3+(20−1)×4 = 3+76 = 79. S₂₀ = 20/2×(3+79) = 10×82 = 820.

Q2MEDIUM· GP

The sum of infinite terms of a GP is 12 and its first term is 4. Find the common ratio and the 4th term.

▶ Show solution

S∞ = a/(1−r) → 12 = 4/(1−r) → 1−r = 1/3 → r = 2/3. Check: |2/3| < 1 ✓. a₄ = 4×(2/3)³ = 4×8/27 = 32/27.

Q3HARD· AM-GM

If a, b, c are in AP and a, b, c are all positive, prove that a + c ≥ 2b, i.e., AM ≥ GM gives the same inequality.

▶ Show solution

Since a, b, c are in AP: b − a = c − b, so 2b = a + c. This means b = (a+c)/2 = AM of a and c. By AM ≥ GM: (a+c)/2 ≥ √(ac), i.e., b ≥ √(ac). So the middle term of an AP is the AM of the two outer terms, and this is ≥ the geometric mean √(ac). Equality holds when a = c.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•AP: aₙ = a+(n−1)d; Sₙ = n/2·[2a+(n−1)d] = n/2·(first + last)
•GP: aₙ = a·rⁿ⁻¹; Sₙ = a(rⁿ−1)/(r−1); S∞ = a/(1−r) only when |r|<1
•AM of a and b = (a+b)/2; GM = √(ab); AM ≥ GM for positive a, b
•To insert n AMs between a and b: d = (b−a)/(n+1); the means are a+d, a+2d, ..., a+nd
•To insert n GMs between a and b: r = (b/a)^(1/(n+1))
•Σk (1 to n) = n(n+1)/2; Σk² = n(n+1)(2n+1)/6; Σk³ = [n(n+1)/2]²
•A sequence is AP iff consecutive differences are constant; GP iff consecutive ratios are constant
•Infinite GP sum exists only when |common ratio| < 1; otherwise the series diverges

CBSE marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 6-8 marks

Question type	Marks each	Typical count	What it tests
Short Answer	2	1	Finding nth term or sum of AP/GP
Long Answer	4-6	1	Special series summation, problems involving both AP and GP, AM-GM applications

Prep strategy

Memorise all four key formulas (AP nth term, AP sum, GP nth term, GP sum) together — they appear together in multi-part questions
For special series, identify the type of expression first: is it linear in k (→Σk), quadratic (→Σk²), or cubic (→Σk³)?
In word problems (salaries, compound interest), translate the context into AP or GP notation before applying any formula

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Simple vs Compound Interest

Simple interest gives amounts in AP (constant addition each year). Compound interest gives amounts in GP (constant multiplication each year) — directly modelled by the GP nth term formula.

Population Growth and Radioactive Decay

Exponential growth (bacteria doubling) and decay (radioactive half-life) are both GP models — each generation multiplies the previous by the same ratio r.

Zeno's Paradox and Infinite Series

Zeno's paradox (can you ever reach a wall if you halve the remaining distance each step?) is resolved by the infinite GP sum: Σ(1/2)ⁿ = 1/(1−1/2) = 2 — a finite answer.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Always identify a, d (for AP) or a, r (for GP) explicitly before substituting into any formula
For 'find the number of terms' questions, set aₙ = last term and solve for n
Special series problems: write out the general term kth as a polynomial in k, then split and apply Σk, Σk², Σk³ individually
In problems about inserting means, find d (or r) first, then list the means systematically

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Arithmetico-Geometric Progression (AGP): if aₙ = (a+(n−1)d)·rⁿ⁻¹, sum using the method of differences
Telescoping series: when terms cancel with adjacent ones, enabling closed-form summation of otherwise complex series

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 11 BoardHigh

JEE MainVery High

JEE AdvancedHigh

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Compute a₂−a₁ and a₃−a₂. If these are equal, it's an AP. Compute a₂/a₁ and a₃/a₂. If these are equal, it's a GP. If neither condition holds, it is neither.

The formula S∞ = a/(1−r) still applies as long as |r| < 1 — including negative r. For example, r = −1/2: the terms alternate in sign but shrink in magnitude, and the sum converges.

For any two positive numbers a and b: AM = (a+b)/2 ≥ √(ab) = GM. Equality holds if and only if a = b.

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