Class 11 Maths Notes

Sequences and Series Class 11 Notes

Complete, exam-ready notes on sequences and series: arithmetic progression (nth term, sum of n terms), geometric progression (nth term, finite and infinite sum), arithmetic and geometric means, the AM–GM inequality, and sum formulas for special series — written for CBSE boards and JEE revision.

Class11SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What are sequences and series about?

Sequences are ordered lists of numbers following a pattern; series are the sums of sequence terms — AP and GP are the two most important types covered at this level.

Sequences and Series — Basics

Sequence

A function whose domain is the set of natural numbers. Each term is denoted a₁, a₂, a₃, ... The pattern or rule of formation determines the sequence.

Series

The sum of the terms of a sequence: $S_n = a_1 + a_2 + \cdots + a_n$ . A series is finite (n terms) or infinite (if the sequence is infinite).

A finite sequence has a fixed number of terms; an infinite sequence continues indefinitely.
A series converges if its partial sums approach a finite limit; otherwise it diverges.
The two most common progressions are arithmetic (AP) and geometric (GP).

Arithmetic Progression (AP)

Arithmetic progression

A sequence where each term differs from the previous by a constant called the common difference d: $a, \; a+d, \; a+2d, \; \ldots$ .

t_n = a + (n-1)d

nth term of AP

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

Sum of n terms of AP

Three numbers in AP can be written as (a − d), a, (a + d) to simplify calculations.
Arithmetic mean of a and b is
$A = \frac{a+b}{2}$
, which is the single term inserted between them to form an AP.

Geometric Progression (GP)

Geometric progression

A sequence where each term is obtained by multiplying the previous by a constant called the common ratio r: $a, \; ar, \; ar^2, \; \ldots$ .

t_n = a \, r^{n-1}

nth term of GP

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

Sum of n terms of GP

S_{\infty} = \frac{a}{1-r}, \quad |r| < 1

Sum to infinity of GP

Three numbers in GP can be written as a/r, a, ar.
Geometric mean of a and b is
$G = \sqrt{ab}$
, which is the single term inserted between them to form a GP.
The sum to infinity exists only when |r| < 1 — the series converges.

Arithmetic Mean and Geometric Mean

AM = \frac{a+b}{2}, \qquad GM = \sqrt{ab}

AM and GM of two numbers

AM ≥ GM inequality

For any non-negative real numbers a and b: $\frac{a+b}{2} \ge \sqrt{ab}$ , with equality if and only if a = b. This inequality is heavily tested in JEE and forms the basis for many optimisation problems.

If AM and GM of two numbers are given, the numbers are the roots of
$x^2 - 2(\text{AM})x + (\text{GM})^2 = 0$
.
For three numbers: AM = (a + b + c)/3 and GM =
$\sqrt[3]{abc}$
.

Sum of Special Series

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

Sum of first n natural numbers

\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}

Sum of first n squares

\sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2

Sum of first n cubes

The sum of cubes equals the square of the sum of the first n natural numbers.
These formulas are used to convert special series into closed-form expressions for quick evaluation.
They are derived by induction or by algebraic manipulation of known identities.

Properties of AP and GP

Adding or subtracting a constant from every term of an AP preserves the common difference but shifts the sum.
Multiplying or dividing every term of an AP by a non-zero constant creates a new AP with the same number of terms but a scaled common difference.
In a GP, multiplying or dividing every term by a non-zero constant preserves the common ratio.
The product of corresponding terms of two APs does not form an AP, but the product of corresponding terms of two GPs with the same ratio is again a GP.
If a, b, c are in AP then 2b = a + c; if a, b, c are in GP then b² = ac.

Quick test for progressions

Three consecutive terms a, b, c form an AP if 2b = a + c. They form a GP if b² = ac (with b ≠ 0). These are the fastest ways to check in MCQs.

Solved Examples

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

Solution: a = 3, d = 4. $t_{20} = 3 + 19 \times 4 = 79$ . $S_{20} = \frac{20}{2}(3 + 79) = 10 \times 82 = 820$ .

Example: The sum of two numbers is 14 and their product is 45. Find the numbers using AM and GM.

Solution: AM = 14/2 = 7, GM = √45 = 3√5. The numbers are the roots of $x^2 - 14x + 45 = 0$ , which factors as (x − 5)(x − 9) = 0. The numbers are 5 and 9.

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

nth term of AP

t_n = a + (n-1)d

Sum of n terms of AP

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

nth term of GP

t_n = a \, r^{n-1}

Sum of n terms of GP

S_n = \frac{a(r^n - 1)}{r - 1}, \; r \neq 1

Sum to infinity of GP

S_{\infty} = \frac{a}{1-r}, \; |r| < 1

AM–GM inequality

AM = \frac{a+b}{2} \ge \sqrt{ab} = GM

Sum of first n squares

\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

AP: t_n = a + (n−1)d and S_n = n/2[2a + (n−1)d] — know these cold.
GP: t_n = ar^{n−1}; sum to infinity exists only when |r| < 1.
AM ≥ GM with equality iff a = b — frequently used in inequalities and optimisation.
Sum of cubes of first n natural numbers = [n(n+1)/2]² — the square of the sum.
Three numbers in AP: (a−d, a, a+d); in GP: (a/r, a, ar) — use symmetry to simplify.
For two numbers, given AM and GM, the numbers are roots of x² − 2(AM)x + (GM)² = 0.

FAQ

Common questions

What is the difference between AP and GP?

In an AP, consecutive terms differ by a constant (common difference d). In a GP, consecutive terms have a constant ratio (common ratio r).

When does the sum of an infinite GP converge?

Only when the common ratio satisfies |r| < 1. The sum to infinity is S∞ = a/(1 − r).

What is the AM–GM inequality?

For any non-negative numbers a and b, (a + b)/2 ≥ √(ab), with equality if and only if a = b. It generalises to more than two numbers.

How do you find the sum of a special series like 1² + 2² + ... + n²?

Use the closed-form formula n(n+1)(2n+1)/6. Similarly, Σk = n(n+1)/2 and Σk³ = [n(n+1)/2]².

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