Sequences and Series Class 11 Notes & Important Questions (Ch 8)

Q: What is in CBSE Class 11 Maths Chapter 8, Sequences and Series?

The topics are: Geometric progressions, Geometric mean and AM ≥ GM, Infinite GP, Arithmetic mean. This page has revision notes for each, and 32 board-style practice questions with step-marked solutions.

Q: How many marks is Sequences and Series worth?

It is in the Algebra unit, which carries 25 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Q: Where do students lose marks in Sequences and Series?

nth term written arⁿ. Fix: aₙ = ar^(n−1). Sum to infinity used with |r| ≥ 1. Fix: check |r| < 1 and say so.

Q: Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

Revision notes

Sequences and Series: revision notes

1. Sequences and series

A sequence is an ordered list $a_{1}, a_{2}, a_{3}, \dots$ , often given by a formula for $a_{n}$ or by a recurrence. A series is the sum of its terms, $a_{1} + a_{2} + \dots$ ; $S_{n}$ is the sum of the first $n$ terms.

2. Arithmetic mean

The AM of $a$ and $b$ is $A = \frac{a + b}{2}$ , so $a, A, b$ are in AP. To insert $n$ AMs between $a$ and $b$ , use common difference $d = \frac{b - a}{n + 1}$ .

3. Geometric progressions

A GP multiplies by a fixed non-zero ratio $r$ : $a, a r, a r^{2}, \dots$

General term: $a_{n} = a r^{n - 1}$ .
Sum of $n$ terms: $S_{n} = \frac{a (r^{n} - 1)}{r - 1} = \frac{a (1 - r^{n})}{1 - r}$ for $r \neq 1$ ; $S_{n} = n a$ if $r = 1$ .
Three terms in GP: $\frac{a}{r}, a, a r$ (their product is $a^{3}$ ).

4. Infinite GP

If $| r | < 1$ , $r^{n} \to 0$ and $S_{\infty} = \frac{a}{1 - r}$ . Recurring decimals are infinite GPs: $0. \overset{―}{36} = \frac{36}{100} + \frac{36}{100^{2}} + \dots = \frac{36}{99} = \frac{4}{11}$ .

5. Geometric mean and AM ≥ GM

The GM of positive $a, b$ is $G = \sqrt{a b}$ . To insert $n$ GMs between $a$ and $b$ , use $r = {(\frac{b}{a})}^{\frac{1}{n + 1}}$ . For positive $a, b$ : $A - G = \frac{(\sqrt{a} - \sqrt{b})^{2}}{2} \geq 0$ , so $A \geq G$ , with equality only when $a = b$ . If $A$ and $G$ are known, $a, b$ are the roots of $x^{2} - 2 A x + G^{2} = 0$ .

Worked example 1

Find the sum of the first $7$ terms of $2, 6, 18, \dots$

$S_{7} = \frac{2 (3^{7} - 1)}{3 - 1} = 3^{7} - 1 = 2186$ .

Worked example 2

Find $1 - \frac{1}{2} + \frac{1}{4} - \dots$ to infinity.

$r = - \frac{1}{2}$ : $S_{\infty} = \frac{1}{1 + \frac{1}{2}} = \frac{2}{3}$ .

Worked example 3

Insert two GMs between $2$ and $54$ .

$54 = 2 r^{3} \Rightarrow r = 3$ : $6, 18$ .

Common errors

$a_{n} = a r^{n}$ instead of $a r^{n - 1}$ .
Using $S_{\infty}$ when $| r | \geq 1$ : the sum does not exist.
Taking the ratio as $\frac{a_{1}}{a_{2}}$ instead of $\frac{a_{2}}{a_{1}}$ , especially with negative ratios.
Forgetting that the GM needs positive numbers.

Exam tips

When two terms of a GP are given, divide the equations to find $r$ first.
In AM–GM questions, state "for positive numbers" before using $A \geq G$ .

Topics in this chapter: Geometric progressions · Geometric mean and AM ≥ GM · Infinite GP · Arithmetic mean.

Formulas for this chapter: Chapter 8 on the Class 11 formula sheet (free PDF and A3 poster).

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can find terms, sums and sums to infinity of GPs and the AM and GM of two numbers.

Read first: 1. Sequences and series; 2. AM; 3. GP; 4. Infinite GP; 5. GM 5 practice questions · checkpoint: 3 questions, 3 marks, pass 80%

Practise step 1

Step 2

Exam standard

You can find a GP from two of its terms, insert means and convert recurring decimals.

Read first: 3. Sum of n terms; 4. Recurring decimals; Worked examples 1-3 6 practice questions · checkpoint: 3 questions, 8 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can sum series built from GPs and model bouncing balls and savings plans.

Read first: 3. GP sums; 4. Infinite GP 3 practice questions · checkpoint: 2 questions, 9 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can use AM ≥ GM and combine conditions on S∞ and partial sums.

Read first: 5. AM ≥ GM; Common errors 3 practice questions · checkpoint: 3 questions, 9 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 3 of the 32 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceGeometric progressions

The $6$ th term of the GP $2, 6, 18, \dots$ is

(a) $486$
(b) $1458$
(c) $162$
(d) $972$

Q2·1 mark·Multiple choiceGeometric mean and AM ≥ GM

The geometric mean of $4$ and $16$ is

(a) $10$
(b) $8$
(c) $64$
(d) $12$

Q3·1 mark·Multiple choiceInfinite GP

The sum to infinity of $1 + \frac{1}{3} + \frac{1}{9} + \dots$ is

(a) $\frac{4}{3}$
(b) $3$
(c) $\frac{3}{2}$
(d) $\frac{2}{3}$

Worked solution·3 marks·Short answerInfinite GP

Write $0. \overset{―}{36}$ as an infinite geometric series and hence as a fraction in lowest terms.

Full worked solution Free, no sign-in

$\frac{36}{100} + \frac{36}{100^{2}} + \dots$ , $a = \frac{36}{100}$ , $r = \frac{1}{100}$ B1
$S_{\infty} = \frac{36 / 100}{1 - 1 / 100} = \frac{36}{99}$ M1
$= \frac{4}{11}$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original algebra problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S93 · Problem S28 · Problem S92 · Problem S27. All 22 →

Knowledge organiser: Sequences and Series on one page (key facts, formulas and mistakes to avoid; printable PDF)

Formulas in this topic: Geometric sequence. All formulas A–Z

Where marks are lost in Sequences and Series

nth term written arⁿ. Fix: aₙ = ar^(n−1).
Sum to infinity used with |r| ≥ 1. Fix: check |r| < 1 and say so.
Two unknowns from two terms solved by subtraction. Fix: divide aₚ by a_q to get r^(p−q).
Negative ratio lost when taking roots (r² = 4 gives r = ±2). Fix: keep both unless the question rules one out.
Recurring decimals converted with the wrong ratio. Fix: one repeating block of length k gives r = 1/10^k.

Test Sequences and Series against the clock

Want it against the clock? Take a timed 30-mark chapter test on Sequences and Series, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Preparing for JEE? Try the JEE-style questions on sequences and series.

Sequences and Series: common questions

What is in CBSE Class 11 Maths Chapter 8, Sequences and Series?