Class 11 Maths NCERT Solutions Chapter 8 Sequences and Series | Boundless Maths
Chapter 8 Class 11 Maths NCERT Solutions · Unit II · Algebra

Class 11 Maths NCERT Solutions Chapter 8: Sequences and Series

Free, step-by-step NCERT Solutions for both exercises of this chapter — writing and evaluating sequences given their general term or a recurrence relation, and geometric progressions: the nth term, the sum to n terms, the geometric mean, and its relationship with the arithmetic mean — plus the Miscellaneous Exercise of mixed applications. Solved the way CBSE awards marks, with the key formulas and the mistakes that cost students marks every year, right on this page.

Sequences and Series builds directly on the Arithmetic Progression students met in Class 10, then introduces its counterpart: the Geometric Progression, where each term is multiplied by a fixed common ratio instead of having a fixed common difference added to it. The chapter's real-world reach is what makes it memorable — compound interest, population growth, machine depreciation and chain letters all turn out to be geometric progressions in disguise, once the first term and common ratio are correctly identified.

3Exercises (incl. Misc.)
64Total Questions
2026-27CBSE Syllabus
100%Solved

Class 11 Maths NCERT Solutions Chapter 8 — Overview

This Class 11 Maths NCERT Solutions Chapter 8 hub covers Sequences and Series, the chapter that formalises what it means for numbers to follow a pattern. A sequence is defined as an ordered list of numbers — formally, a function whose domain is the natural numbers — and its terms may be given directly by a formula for the nth term, or indirectly through a recurrence relation, where each term is built from the ones before it, as in the famous Fibonacci sequence. A series is simply the sum of a sequence's terms, often written compactly using sigma notation.

The chapter's main focus is the Geometric Progression (G.P.) — a sequence where every term is a fixed multiple, the common ratio, of the one before it. Once the nth-term formula and the sum-to-n-terms formula are in hand, the chapter turns to the geometric mean of two numbers, and finally to the relationship between the arithmetic mean (A.M.) and geometric mean (G.M.), which turns out to always satisfy A ≥ G. The Miscellaneous Exercise closes the chapter with longer, applied problems — compound interest, depreciating machinery, chain letters and instalment payments — which are really G.P. problems wearing a real-world costume.

How the Chapter Builds

One Idea Leads to the Next

1

Sequences & Series

General term, recurrence relations, and sigma notation for series. Exercise 8.1.

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2

Geometric Progression

The nth term aₙ = arⁿ⁻¹ and the sum to n terms of a G.P. Exercise 8.2.

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3

Geometric Mean

G = √(ab), and inserting several geometric means between two numbers. Exercise 8.2.

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4

A.M.–G.M. Relationship

Proving A ≥ G, and recovering two numbers from their A.M. and G.M. Exercise 8.2.

Quick Reference

Important Formulas — Chapter 8

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below has the full printable version for all of Sequences and Series.

Sequences and Series (§8.2–8.3)

General term of a sequence

a_n

The number at the nth position of the sequence, either given directly by a formula in n, or built up from earlier terms.

Recurrence relation

a_1=1,\ a_n=a_{n-1}+2

Each term is defined using the term(s) before it — the Fibonacci sequence is the classic example.

Series (sigma notation)

a_1+a_2+\cdots+a_n=\sum_{k=1}^{n}a_k

The indicated sum of a sequence's terms — "series" refers to the sum written out, not the number it adds up to.

Geometric Progression (§8.4)

nth term of a G.P.

a_n=ar^{n-1}

a is the first term, r is the common ratio — every term is r times the one before it.

Sum to n terms

S_n=\dfrac{a(r^n-1)}{r-1},\ r\ne1 \qquad S_n=na,\ r=1

Always check whether r = 1 first — the standard formula divides by zero in that case.

Identifying a G.P.

\dfrac{a_{k+1}}{a_k}=r \text{ (constant)}

A sequence is a G.P. exactly when the ratio of every term to the one before it is the same fixed number.

Geometric Mean & A.M.–G.M. Relationship (§8.4.3–8.5)

Geometric mean of two numbers

G=\sqrt{ab}

Defined for two positive numbers a and b — a, G, b then form 3 consecutive terms of a G.P.

Inserting n geometric means

G_k=a\left(\dfrac{b}{a}\right)^{\frac{k}{n+1}}

Treats a, G₁, G₂, …, Gₙ, b as an (n+2)-term G.P., and solves for the common ratio.

A.M.–G.M. relationship

A=\dfrac{a+b}{2},\quad G=\sqrt{ab},\quad A\ge G

Equality holds only when a = b, since A − G equals half of (√a − √b)², which is never negative.

Decision Guide

Which Sequences & Series Concept Applies?

A quick way to decide, once you know what the question is actually asking for.

What the question is askingUse thisWhy
Write out terms given a formula for the nth termDirect substitutionSubstitute n = 1, 2, 3, … into the given formula for aₙ (§8.2).
Write out terms given a rule linking each term to earlier onesRecurrence relationBuild up terms one at a time, starting from the given first term(s) (§8.2).
Decide whether a sequence is geometricCheck the common ratioConfirm every term divided by the one before it gives the same constant r (§8.4).
Find a specific term of a G.P.aₙ = arⁿ⁻¹Only needs the first term, common ratio, and position n (§8.4.1).
Find the total of the first n terms of a G.P.Sₙ formulaUse the r ≠ 1 case unless the common ratio is exactly 1 (§8.4.2).
Insert numbers between a and b to form a G.P.Geometric mean(s)Treat the full list as one G.P. and solve for the common ratio (§8.4.3).
Given the A.M. and G.M. of two numbers, find the numbersA.M.–G.M. systemUse a + b = 2A and ab = G² together, then solve the resulting quadratic (§8.5).
A series like 7, 77, 777, ... that isn't quite a G.P.Relate it to a G.P. algebraicallyFactor out a constant and rewrite each term as a power of 10 minus 1 (Misc. Exercise).
Compound interest, depreciation, or a chain-letter problemRecognise the hidden G.P.Identify the first term and the growth/decay factor as a and r (Misc. Exercise).
Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across both exercises.

  • Forgetting the r = 1 special case — the standard sum formula Sₙ = a(rⁿ−1)/(r−1) divides by zero when r = 1; in that case the sum is simply Sₙ = na.
  • Confusing common difference with common ratio — an A.P. adds a fixed number to get the next term, a G.P. multiplies by one; mixing the two formulas up is a frequent slip right after finishing the A.P. chapter.
  • Losing the sign when the common ratio is negative — a negative r, such as −3/4, makes terms alternate in sign; dropping a negative sign partway through a multi-step problem is easy to do and hard to spot later.
  • Forgetting the geometric mean needs positive numbers — G = √(ab) is only defined this way for a, b > 0; students sometimes apply it without checking the sign of the numbers involved.
  • Assuming A and G can be equal in general — A ≥ G always holds for positive numbers, with equality only when a = b; treating A = G as a general fact rather than a special case is a common misconception.
  • Building a recurrence-relation sequence incorrectly — miscalculating even one early term (a₂, a₃, …) throws off every term that follows it, since each one depends on the ones before.
  • Not identifying a, r and n correctly in worded applications — in compound interest, depreciation or chain-letter problems, misreading which quantity is the first term, which is the common ratio, and how many terms are involved is the single biggest source of wrong answers in the Miscellaneous Exercise.
Solve Chapter-Wise

Class 11 Maths NCERT Solutions Chapter 8 — Choose an Exercise

8.1

Exercise 8.1

Sequences — general term, recurrence relations, and writing out the corresponding series · 14 questions

Solve Exercise 8.1 →
8.2

Exercise 8.2

Geometric Progression — nth term, sum to n terms, geometric mean, and the A.M.–G.M. relationship · 32 questions

Solve Exercise 8.2 →
M

Miscellaneous Exercise

Mixed problems and real-world applications — compound growth, depreciation, chain letters and instalments · 18 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for Sequences and Series — plus every other Class 11 Maths chapter — in one printable PDF.

Get Formula Cards →
Common Questions

Frequently Asked Questions

Quick answers from Class 11 Maths NCERT Solutions Chapter 8, Sequences and Series.

How many exercises are there in Chapter 8, Sequences and Series?
There are two main exercises — 8.1 (Sequences, 14 questions) and 8.2 (Geometric Progression, 32 questions) — plus a Miscellaneous Exercise of 18 questions covering mixed applications, totalling 64 questions.
What is a Geometric Progression (G.P.)?
A Geometric Progression is a sequence in which every term, except the first, is obtained by multiplying the preceding term by a fixed non-zero number called the common ratio. If the first term is a and the common ratio is r, the sequence is a, ar, ar², ar³, ... and the nth term is given by aₙ = arⁿ⁻¹.
What is the relationship between the arithmetic mean (A.M.) and geometric mean (G.M.) of two positive numbers?
For two positive real numbers a and b, the arithmetic mean is A = (a + b)/2 and the geometric mean is G = √(ab). It can be shown that A is always greater than or equal to G, with equality holding only when a equals b. This follows since A − G equals half of (√a − √b)², which can never be negative.
How is the sum of a G.P. different from the nth term of a G.P.?
The nth term, aₙ = arⁿ⁻¹, gives the value of one specific term of the progression. The sum to n terms, denoted Sₙ, adds up all the terms from the first to the nth: Sₙ = a(rⁿ − 1)/(r − 1) when the common ratio r is not equal to 1, and Sₙ = na in the special case r = 1.
Where can I find the official NCERT textbook for this chapter?
Sequences and Series is Chapter 8 of the NCERT Class 11 Mathematics textbook, published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this page follow the exercises exactly as they appear there.
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