Class 11 Maths Chapter 8: Sequences and Series

Sequences and Series is Chapter 8 of NCERT Class 11 Maths. It studies ordered lists of numbers that follow a rule, with special focus on arithmetic progressions, geometric progressions, their means, and the useful relation between the arithmetic mean and the geometric mean. Watch the 2 video lessons below, then use the notes, formulas and solved examples to revise for your CBSE exams.

What you will learn

The meaning of a sequence, a series and the general term
A quick recap of arithmetic progression (AP) and the arithmetic mean
The general term and sum of n terms of a geometric progression (GP)
The sum of an infinite GP and how to use it for recurring decimals
Geometric mean and the relation AM ≥ GM

Video Lessons

Sequences and Series - Geometric Progression Revision

Sequences and Series - Geometric Progression Extra Questions

Key Concepts and Formulas

Sequence and Series

A sequence is a list of numbers arranged in a definite order according to a rule. Each number is called a term, and the nth term is written aₙ. When we add the terms of a sequence, we get a series.

Series: a₁ + a₂ + a₃ + ... + aₙ
Sum of n terms: Sₙ

A sequence with a limited number of terms is finite; otherwise it is infinite
The rule for the nth term, aₙ, is called the general term
Some sequences, like the Fibonacci sequence, are defined by a recurrence relation linking each term to earlier ones

Arithmetic Progression Recap

In an AP every term differs from the previous one by a fixed number d, called the common difference. You met this in Class 10, and the formulas are used here as a base for GP.

aₙ = a + (n − 1)d
Sₙ = (n/2)(2a + (n − 1)d)
Sₙ = (n/2)(a + l),  l = last term
AM of a and b: A = (a + b)/2
n AMs between a and b: d = (b − a)/(n + 1)

Geometric Progression

In a GP every term is obtained by multiplying the previous term by a fixed non-zero number r, called the common ratio. The terms grow or shrink by a constant factor.

Standard form: a, ar, ar², ar³, ...
aₙ = arⁿ⁻¹
r = aₙ₊₁ / aₙ
Three terms in GP: a/r, a, ar

Bar chart of the geometric progression 2, 6, 18, 54, 162 for Class 11 Sequences and Series — Each term is three times the previous one

Taking three terms as a/r, a, ar makes their product simply a³
If every term of a GP is multiplied by the same non-zero constant, or replaced by its reciprocal, the result is still a GP

Sum of a GP

The sum formula is derived by writing Sₙ, multiplying it by r, and subtracting. Learn when to use each version.

Sₙ = a(rⁿ − 1)/(r − 1)   (convenient when r > 1)
Sₙ = a(1 − rⁿ)/(1 − r)   (convenient when |r| < 1)
If r = 1: Sₙ = na
Infinite GP, |r| < 1: S∞ = a/(1 − r)

If |r| ≥ 1, the sum of an infinite GP does not exist
Recurring decimals such as 0.272727... can be written as an infinite GP and converted into a fraction

Geometric Mean and AM GM Relation

The geometric mean of two positive numbers is the middle term that makes the three numbers a GP.

GM of a and b: G = √(ab)
n GMs between a and b: b = arⁿ⁺¹
A − G = (√a − √b)²/2 ≥ 0
So AM ≥ GM, with A = G only when a = b
Numbers from A and G: x² − 2Ax + G² = 0

Solved Examples

Example 1: Find the sum of the first 20 terms of the AP 7, 11, 15, ...

a = 7, d = 4, n = 20
S₂₀ = (20/2)(2 × 7 + 19 × 4)
= 10 × (14 + 76)
= 10 × 90
∴ S₂₀ = 900

Example 2: Find the 9th term of the GP 3, −6, 12, ...

a = 3, r = −6/3 = −2
a₉ = 3 × (−2)⁸
= 3 × 256
∴ a₉ = 768

Example 3: Which term of the GP 5, 15, 45, ... is 3645?

a = 5, r = 3
5 × 3ⁿ⁻¹ = 3645
3ⁿ⁻¹ = 729 = 3⁶
n − 1 = 6
∴ 3645 is the 7th term

Example 4: Find the sum of the first 8 terms of the GP 2, 6, 18, ...

a = 2, r = 3, n = 8
S₈ = 2(3⁸ − 1)/(3 − 1)
= 3⁸ − 1
= 6561 − 1
∴ S₈ = 6560

Example 5: Find the sum of the infinite GP 12 + 4 + 4/3 + ...

a = 12, r = 4/12 = 1/3, |r| < 1
S∞ = 12/(1 − 1/3)
= 12 ÷ (2/3)
∴ S∞ = 18

Bar showing the infinite GP 12 + 4 + 4/3 adding up towards 18, Class 11 Sequences and Series — The partial sums get closer and closer to 18

Example 6: Insert three geometric means between 2 and 162.

Total terms = 5, so 162 = 2 × r⁴
r⁴ = 81
r = 3 (taking the positive ratio)
Terms: 2, 6, 18, 54, 162
∴ GMs are 6, 18 and 54

Example 7: The AM and GM of two positive numbers are 15 and 12. Find the numbers.

a + b = 2 × 15 = 30
ab = 12² = 144
x² − 30x + 144 = 0
(x − 24)(x − 6) = 0
∴ The numbers are 24 and 6

Common Mistakes to Avoid

Writing aₙ = arⁿ instead of arⁿ⁻¹. The first term already uses r⁰.
Using the infinite sum formula when |r| ≥ 1. Always check |r| < 1 first.
Forgetting that the sign of r matters. With a negative ratio the terms alternate in sign, so be careful with even and odd powers.
When inserting n means, remember the total number of terms becomes n + 2, so the power of r is n + 1.

Quick Revision

AP: aₙ = a + (n − 1)d
AP: Sₙ = (n/2)(2a + (n − 1)d)
AM = (a + b)/2
GP: aₙ = arⁿ⁻¹
GP: Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1
GP: Sₙ = na, r = 1
S∞ = a/(1 − r), only when |r| < 1
GM = √(ab),  AM ≥ GM