📊 CBSE · Class 11 · Mathematics · Chapter 9

Sequences and
Series

Complete chapter resources for CBSE Class 11 Mathematics — topic breakdown, key formulas for AP, GP, and special series, sample questions, previous year board questions, and instant AI question paper generation.

4Topics
6–8Board marks
8Sample questions
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Key Formulas — Chapter 9
  • AP nth term: a_n = a + (n−1)d
  • AP sum (Sn): Sn = n/2 × [2a + (n−1)d]
  • GP nth term: a_n = a · r^(n−1)
  • GP sum (Sn): Sn = a(r^n − 1) / (r − 1), r ≠ 1
  • Infinite GP: S∞ = a / (1 − r), |r| < 1
  • Σn² = n(n+1)(2n+1) / 6
Chapter Overview

What this chapter covers

A sequence is an ordered list of numbers following a rule, and a series is the sum of the terms of a sequence. Chapter 9 of NCERT Class 11 Mathematics introduces two fundamental types of sequences: the Arithmetic Progression (AP), where each term differs from the previous by a constant value called the common difference d, and the Geometric Progression (GP), where each term is obtained by multiplying the previous term by a constant called the common ratio r.

For each progression the chapter derives the general (nth) term formula and the formula for the sum of the first n terms. A key concept in AP is the insertion of arithmetic means between two numbers, and in GP the insertion of geometric means. The chapter also establishes the relationship between AM and GM — specifically the inequality AM ≥ GM — which is widely used in advanced algebra and optimization problems.

The final section covers special series: the sum of the first n natural numbers (Σn), the sum of their squares (Σn²), and the sum of their cubes (Σn³). Board questions frequently ask students to simplify or evaluate expressions like Σ(3k² + 2k − 1) by applying these standard results, making memorisation of all three formulas essential for scoring well.

Topics

What's inside Chapter 9

As per NCERT Class 11 Mathematics (CBSE syllabus)

Topic 1
Arithmetic Progressions (AP)
Definition, common difference, nth term formula a_n = a + (n−1)d, sum formula Sn = n/2[2a + (n−1)d], insertion of arithmetic means, and properties of AP.
Topic 2
Geometric Progressions (GP)
Definition, common ratio, nth term a_n = ar^(n−1), sum of n terms, infinite GP sum S∞ = a/(1−r) for |r| < 1, and insertion of geometric means.
Topic 3
Relationship between AM and GM
Proof that AM ≥ GM for positive real numbers. AM = (a+b)/2, GM = √(ab). Applications in finding conditions under which equality holds and in optimization problems.
Topic 4
Sum to n Terms of Special Series
Formulas for Σn = n(n+1)/2, Σn² = n(n+1)(2n+1)/6, and Σn³ = [n(n+1)/2]². Evaluating sums of series by decomposing into these standard forms.
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Class 10 · Arithmetic Progressions
nth term and sum formulas introduced in Class 10 Ch 5
Ch 5 · Binomial Theorem
Sigma notation and combinatorial sums used across chapters
Chapter 9 Sequences
& Series
Leads to
Class 12 · Limits & Calculus
Infinite series concepts underpin limits and convergence
JEE / Competitive Maths
AGP, telescoping series, and advanced summation techniques
Board Blueprint

Marks & question-type breakdown

Typical pattern based on CBSE Class 11 Mathematics board papers from the last five years.

Question type Marks Typical count What's usually tested
MCQ / Objective 1 1–2 Identify AP/GP, find common difference or ratio, nature of sequence
Very Short Answer 2 1 Find the nth term of AP/GP, insert means, or verify AM ≥ GM
Short Answer 3 1 Sum of n terms of AP or GP, sum of infinite GP, or special series evaluation
Long Answer / Word Problem 4–5 1 Real-life AP/GP application, inserting multiple means, or multi-part special series
Total (approximate) 6–8 4–5 Weightage varies across paper sets and years
Sample Questions

8 sample questions — generated by MarksZen AI

Aligned to CBSE Class 11 Mathematics Chapter 9. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
The 10th term of the AP 3, 7, 11, 15, … is: (a) 39 (b) 40 (c) 43 (d) 41
Q2 Easy 2 marks Short Answer
Find the sum of the first 20 terms of the AP: 5, 8, 11, 14, …
Q3 Medium 2 marks Short Answer
The first term of a GP is 2 and its common ratio is 3. Find the 6th term and the sum of the first 6 terms.
Q4 Medium 3 marks Short Answer
Find the sum of the infinite GP: 1/2 + 1/4 + 1/8 + 1/16 + … Verify that the common ratio satisfies the condition for convergence.
Q5 Medium 3 marks Short Answer
Insert three arithmetic means between 5 and 21. Verify that the five numbers form an AP and find their sum using the AP sum formula.
Q6 Hard 4 marks Short Answer
Find the sum: Σ(k=1 to n) of (2k² + 3k − 1) using the standard results for Σk and Σk². Simplify your answer fully in terms of n.
Q7 Hard 4 marks Word Problem
A person saves ₹500 in the first month and increases savings by ₹100 each subsequent month. (i) Write the sequence of monthly savings as an AP. (ii) Find the total savings at the end of 24 months. (iii) In which month do the savings first exceed ₹2000?
Q8 Hard 5 marks Word Problem
A ball is dropped from a height of 64 m. Each time it hits the ground, it bounces back to three-quarters of the height from which it fell. (i) Show that the heights of successive bounces form a GP and find the common ratio. (ii) Find the total distance travelled by the ball before it comes to rest. (iii) Find the height of the 5th bounce.
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Previous Year Questions

From CBSE board examinations

Actual questions from past Class 11 Mathematics board papers — Sequences and Series chapter.

Board 20234 marks
If the sum of first p terms of an AP is the same as the sum of its first q terms (p ≠ q), show that the sum of its first (p + q) terms is zero. (CBSE All India 2023)
Board 20223 marks
Find the sum of the series: 5 + 55 + 555 + 5555 + … to n terms. (CBSE Delhi 2022)
Board 20204 marks
If A and G are respectively the AM and GM between two positive numbers a and b, prove that a and b are the roots of the equation x² − 2Ax + G² = 0. (CBSE 2020)
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FAQ

Questions teachers ask

How many marks does Sequences and Series carry in the CBSE Class 11 Mathematics exam? +
Sequences and Series typically carries 6–8 marks in the CBSE Class 11 Mathematics year-end examination. Questions are spread across 1-mark MCQs (nature of a sequence or series), 2-mark short answers (finding the nth term or sum), and 4–5 mark long answers involving word problems, insertion of means, or special series.
What is the difference between an AP and a GP, and how should students decide which formula to use? +
An Arithmetic Progression (AP) has a constant difference between consecutive terms, while a Geometric Progression (GP) has a constant ratio. To decide: if consecutive terms differ by the same value, use AP formulas (nth term = a + (n−1)d; sum Sn = n/2 × [2a + (n−1)d]). If consecutive terms have the same ratio, use GP formulas (nth term = ar^(n−1); sum Sn = a(r^n − 1)/(r − 1) for r ≠ 1).
What are the special series formulas that students must memorise for board exams? +
Three special series sums are directly examinable: (1) Sum of first n natural numbers: Σn = n(n+1)/2. (2) Sum of squares of first n natural numbers: Σn² = n(n+1)(2n+1)/6. (3) Sum of cubes of first n natural numbers: Σn³ = [n(n+1)/2]². Board questions often ask students to find the sum of a series like Σ(2k² + 3k) by splitting and applying these formulas.
What is the sum of an infinite GP and when does it exist? +
The sum of an infinite GP S∞ = a / (1 − r) exists only when |r| < 1, i.e., the common ratio is between −1 and 1 (exclusive). When |r| ≥ 1, the terms do not approach zero and the series diverges. This is a common 2-mark board question — students must first verify |r| < 1 before applying the formula.
How do I generate a custom question paper for Sequences and Series using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 11 Mathematics, select Chapter 9 (Sequences and Series), set your preferred question-type mix and total marks — the AI generates a complete board-aligned paper with answer key in under 2 minutes, ready for PDF export.