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Mathematics Grade XI

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Chapter 8: Sequences and Series

Class 11 • Mathematics • Chapter 8

Sequences and Series

Numbers that follow a clear pattern, and the neat formulae for adding them up.

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Chapter Roadmap

Sequences • Series • Arithmetic Progression • Geometric Progression • Sum of a GP • Arithmetic and Geometric Means • Key Results • Applications

Why Sequences and Series Matter

Patterns of numbers are everywhere. The instalments of a loan, the bounces of a ball as it loses height, the way savings grow with interest, and even the spirals of a sunflower all follow ordered lists of numbers. A sequence is simply such an ordered list, and a series is what you get when you add the terms of a sequence together.

Adding a long list one number at a time is slow and error-prone, so mathematics gives us short formulae that find the sum in a single step. In this chapter we study two of the most important patterns: the arithmetic progression, where each term increases by a fixed amount, and the geometric progression, where each term is multiplied by a fixed number. We also meet the arithmetic mean and the geometric mean, and a striking relationship between them that holds for all positive numbers.

Key idea

In an arithmetic progression you add the same number d each time. In a geometric progression you multiply by the same number r each time. Spotting which pattern a list follows is the first step in every problem.

Key Terms You Must Know

Term

Meaning

Example

Sequence

An ordered list of numbers following a rule.

2, 4, 6, 8, …

Term

Each individual number in a sequence, written aₙ for the nth term.

a₃ is the third term

Series

The sum of the terms of a sequence.

2 + 4 + 6 + 8

Arithmetic Progression

A sequence with a constant difference d between terms.

3, 7, 11, 15 (d = 4)

Geometric Progression

A sequence with a constant ratio r between terms.

2, 6, 18, 54 (r = 3)

Arithmetic Mean

The AM of a and b is (a + b)/2.

AM of 4 and 16 is 10

Geometric Mean

The GM of positive a and b is √(ab).

GM of 4 and 16 is 8

Core Concepts, Step by Step

1. Sequences and Series

A sequence is a list of numbers written in a definite order, such as 2, 4, 6, 8. Each number is a term, and we write the nth term as aₙ. A sequence may be finite (it stops) or infinite (it continues without end, shown with three dots). When we add the terms of a sequence we form a series, for example 2 + 4 + 6 + 8. The whole chapter is about finding such sums quickly.

2. Arithmetic Progression (AP)

An arithmetic progression is a sequence in which each term is obtained by adding a fixed number, the common difference d, to the term before it. If the first term is a, the terms are a, a + d, a + 2d, and so on. The nth term is aₙ = a + (n − 1)d. To check whether a sequence is an AP, subtract each term from the next: if the difference is always the same, it is an AP.

3. Geometric Progression (GP)

A geometric progression is a sequence in which each term is obtained by multiplying the term before it by a fixed number, the common ratio r. If the first term is a, the terms are a, ar, ar², and so on. The nth term is aₙ = a rⁿ₋¹. To check for a GP, divide each term by the one before: if the ratio is always the same, it is a GP. The table below sets the two patterns side by side.

Arithmetic progression or geometric progression?

Feature

Arithmetic Progression

Geometric Progression

How it grows

Add the same d

Multiply by the same r

Example

3, 7, 11, 15

2, 6, 18, 54

nth term

aₙ = a + (n − 1)d

aₙ = a rⁿ₋¹

Sum of n terms

Sₙ = (n/2)[2a + (n − 1)d]

Sₙ = a(rⁿ − 1)/(r − 1)

4. The Sum of a Geometric Progression

The sum of the first n terms of a GP with first term a and ratio r (where r is not 1) is Sₙ = a(rⁿ − 1)/(r − 1). Something remarkable happens for an infinite GP when the ratio lies strictly between −1 and 1. The terms shrink towards zero, and the whole infinite sum settles down to the finite value S = a/(1 − r). This is how an unending list of numbers can still add up to something finite, such as 1 + ½ + ¼ + … = 2.

5. Arithmetic and Geometric Means

Between any two numbers we can insert a mean. The arithmetic mean (AM) of a and b is (a + b)/2; inserting it makes a, AM, b into an AP. The geometric mean (GM) of two positive numbers a and b is √(ab); inserting it makes a, GM, b into a GP. You can also insert several means between two numbers, which simply creates a longer AP or GP.

6. The Relationship Between AM and GM

For any two positive numbers, the arithmetic mean is always greater than or equal to the geometric mean: AM ≥ GM. The two are equal only when the numbers are the same. For example, for 4 and 16 the AM is 10 and the GM is 8, and indeed 10 ≥ 8. This simple inequality is one of the most useful tools in all of mathematics, especially for finding greatest and least values.

Key Results & Proofs

Three results do almost all the work in this chapter. Each is short to prove and well worth understanding.

Result 1: Sum of an Arithmetic Progression

Statement. The sum of the first n terms of an AP is Sₙ = (n/2)[2a + (n − 1)d].

Proof

The clever trick is to write the sum forwards and backwards, then add.

S

=

a + (a+d) + … + (l)

write the sum, where l is the last term

S

=

l + (l−d) + … + (a)

write it again, in reverse order

2S

=

(a + l) + (a + l) + … + (a + l)

add the two lines term by term

2S

=

n(a + l)

there are n equal brackets

S

=

(n/2)(a + l)

divide by 2

Sₙ

=

(n/2)[2a + (n − 1)d]

since the last term l = a + (n − 1)d

This pairing trick is the same one the young Gauss is said to have used to add 1 to 100 in seconds.

Result 2: Sum of a Geometric Progression

Statement. The sum of the first n terms of a GP (with r not equal to 1) is Sₙ = a(rⁿ − 1)/(r − 1).

Proof

Multiply the whole sum by r, then subtract, so nearly everything cancels.

Sₙ

=

a + ar + ar² + … + arⁿ₋¹

write the sum

rSₙ

=

ar + ar² + … + arⁿ

multiply every term by r

Sₙ − rSₙ

=

a − arⁿ

subtract; the middle terms cancel

Sₙ(1 − r)

=

a(1 − rⁿ)

factorise both sides

Sₙ

=

a(rⁿ − 1)/(r − 1)

divide and tidy the signs

When −1 < r < 1 and n grows without limit, rⁿ tends to 0, giving the infinite sum S = a/(1 − r).

Result 3: AM is Never Less Than GM

Statement. For any two positive numbers a and b, (a + b)/2 ≥ √(ab).

Proof

Everything follows from one fact: a square can never be negative.

For any real numbers, a square is never negative: (√a − √b)² ≥ 0. – start from a perfect square

a − 2√(ab) + b

≥

0

expand the square

a + b

≥

2√(ab)

move the middle term across

(a + b)/2

≥

√(ab)

divide both sides by 2

Equality holds only when √a = √b, that is when a = b. So AM = GM exactly when the two numbers are equal.

Worked Examples

Example 1

Question: Find the 10th term of the AP 3, 7, 11, …

▶ Show full working

Identify a and d, then use aₙ = a + (n − 1)d.

First term a = 3, common difference d = 7 − 3 = 4. – read off a and d

a₁₀

=

3 + (10 − 1)(4)

substitute n = 10

a₁₀

=

3 + 36 = 39

simplify

Answer: The 10th term is 39.

Example 2

Question: Find the sum of the first 20 terms of the AP 2, 5, 8, …

▶ Show full working

Here a = 2 and d = 3. Use Sₙ = (n/2)[2a + (n − 1)d].

S₂₀

=

(20/2)[2(2) + (20 − 1)(3)]

substitute n = 20

S₂₀

=

10[4 + 57]

work inside the bracket

S₂₀

=

10 × 61 = 610

multiply

Answer: The sum is 610.

Example 3

Question: Which term of the AP 5, 8, 11, … is equal to 92?

▶ Show full working

Set aₙ = 92 with a = 5 and d = 3, then solve for n.

5 + (n − 1)(3)

=

92

set the nth term equal to 92

(n − 1)(3)

=

87

subtract 5

n − 1

=

29

divide by 3

n

=

30

add 1

Answer: 92 is the 30th term.

Example 4

Question: Find the 6th term of the GP 2, 6, 18, …

▶ Show full working

Here a = 2 and r = 6 ÷ 2 = 3. Use aₙ = a rⁿ₋¹.

a₆

=

2 × 3^(6−1)

substitute n = 6

a₆

=

2 × 3^5

simplify the power

a₆

=

2 × 243 = 486

evaluate

Answer: The 6th term is 486.

Example 5

Question: Find the sum of the first 5 terms of the GP 3, 6, 12, …

▶ Show full working

Here a = 3 and r = 2. Use Sₙ = a(rⁿ − 1)/(r − 1).

S₅

=

3(2^5 − 1)/(2 − 1)

substitute n = 5

S₅

=

3(32 − 1)/1

evaluate the power

S₅

=

3 × 31 = 93

simplify

Answer: The sum is 93.

Example 6

Question: Find the sum of the infinite GP 1 + ½ + ¼ + …

▶ Show full working

Here a = 1 and r = ½, and since −1 < r < 1 the infinite sum is finite.

S

=

a/(1 − r)

infinite GP formula

S

=

1/(1 − ½)

substitute a = 1, r = ½

S

=

1/(½) = 2

simplify

Answer: The infinite sum is 2.

Example 7

Question: Insert the arithmetic mean between 4 and 16.

▶ Show full working

The AM of two numbers is simply their average.

AM

=

(4 + 16)/2

arithmetic mean formula

AM

=

20/2 = 10

simplify

Answer: The arithmetic mean is 10, and 4, 10, 16 form an AP.

Example 8

Question: Find the geometric mean between 4 and 16.

▶ Show full working

The GM of two positive numbers is the square root of their product.

GM

=

√(4 × 16)

geometric mean formula

GM

=

√64 = 8

simplify

Answer: The geometric mean is 8, and 4, 8, 16 form a GP.

Example 9

Question: Find the sum 1 + 2 + 3 + … + 100.

▶ Show full working

This is an AP with a = 1, last term l = 100 and n = 100. Use S = (n/2)(a + l).

S

=

(100/2)(1 + 100)

substitute into the sum formula

S

=

50 × 101

simplify

S

=

5050

multiply

Answer: The sum is 5050.

Example 10

Question: The 3rd term of a GP is 24 and the 6th term is 192. Find the first term and the common ratio.

▶ Show full working

Write both terms with the formula a rⁿ₋¹, then divide one equation by the other to remove a.

a r²

=

24

third term

a r⁵

=

192

sixth term

r³

=

192/24 = 8

divide the second by the first

r

=

2

cube root of 8

a

=

24/r² = 24/4 = 6

substitute back

Answer: First term 6 and common ratio 2, so the GP is 6, 12, 24, …

Example 11

Question: A person saves 200 rupees in the first month, 250 in the second, 300 in the third, and so on. How much is saved in 12 months?

▶ Show full working

The savings form an AP with a = 200 and d = 50. Find the sum of 12 terms.

S₁₂

=

(12/2)[2(200) + (12 − 1)(50)]

sum of 12 terms

S₁₂

=

6[400 + 550]

work inside the bracket

S₁₂

=

6 × 950 = 5700

multiply

Answer: A total of 5,700 rupees is saved in 12 months.

Example 12

Question: Verify that AM ≥ GM for the numbers 4 and 9.

▶ Show full working

Work out both means and compare them.

AM

=

(4 + 9)/2 = 6.5

arithmetic mean

GM

=

√(4 × 9) = √36 = 6

geometric mean

Since 6.5 ≥ 6, the inequality AM ≥ GM holds. – compare

Answer: AM = 6.5 and GM = 6, so AM ≥ GM is confirmed.

Where You Meet This in Real Life

Savings and loans

Equal monthly instalments form an arithmetic progression, while money growing at a fixed interest rate each year forms a geometric progression.

Population and growth

Populations, bacteria and viral shares often grow by a fixed multiple in each period, which is exactly a geometric progression.

Bouncing and decay

A ball that rebounds to a fixed fraction of its height, or a medicine that leaves the body at a steady rate, follows a decreasing GP.

Patterns in nature

The Fibonacci sequence and related ratios appear in petals, shells and pinecones, linking sequences to the world around us.

Salary and increments

A starting salary with a fixed yearly increment is an AP, and the total earned over a career is the sum of that AP.

Practice Sets A to D

Practice Set A – Basics

A1. Find the 12th term of the AP 1, 4, 7, …

▶ Reveal full working

Use aₙ = a + (n − 1)d with a = 1, d = 3.

a₁₂

=

1 + (12 − 1)(3)

substitute

a₁₂

=

1 + 33 = 34

simplify

Answer: 34.

A2. Find the sum of the first 10 terms of the AP 5, 10, 15, …

▶ Reveal full working

Here a = 5, d = 5. Use the sum formula.

S₁₀

=

(10/2)[2(5) + 9(5)]

substitute

S₁₀

=

5[10 + 45] = 5 × 55 = 275

simplify

Answer: 275.

A3. Find the 5th term of the GP 1, 3, 9, …

▶ Reveal full working

Here a = 1, r = 3. Use aₙ = a rⁿ₋¹.

a₅

=

1 × 3^4

substitute n = 5

a₅

=

81

evaluate

Answer: 81.

A4. Find the arithmetic mean of 10 and 20.

▶ Reveal full working

Average the two numbers.

AM

=

(10 + 20)/2 = 15

AM formula

Answer: 15.

Practice Set B – Conceptual

B1. What is the difference between a sequence and a series?

▶ Reveal full working

Think about ordering versus adding.

A sequence is an ordered list of numbers.

A series is the sum of the terms of a sequence.

Answer: A sequence is a list; a series is the sum of that list.

B2. When does an infinite geometric progression have a finite sum?

▶ Reveal full working

Look at the size of the common ratio.

The terms must shrink towards zero.

This happens only when −1 < r < 1.

Answer: Only when the common ratio satisfies −1 < r < 1.

B3. Is the sequence 1, 1, 1, 1 an AP, a GP, or both?

▶ Reveal full working

Check the common difference and the common ratio.

The difference between terms is 0, so it is an AP with d = 0.

The ratio between terms is 1, so it is also a GP with r = 1.

Answer: It is both: an AP with d = 0 and a GP with r = 1.

B4. For two positive numbers, when is the AM equal to the GM?

▶ Reveal full working

Recall the equality condition of AM ≥ GM.

Equality in (√a − √b)² ≥ 0 needs √a = √b.

So AM = GM only when a = b.

Answer: Only when the two numbers are equal.

Practice Set C – Application / Numerical

C1. Which term of the GP 3, 6, 12, … is 384?

▶ Reveal full working

Set a rⁿ₋¹ = 384 with a = 3, r = 2.

3 × 2^(n−1)

=

384

set the nth term equal to 384

2^(n−1)

=

128

divide by 3

n − 1

=

7

since 2^7 = 128

n

=

8

add 1

Answer: It is the 8th term.

C2. Find the sum of the infinite GP 4 + 4/3 + 4/9 + …

▶ Reveal full working

Here a = 4 and r = 1/3, so use S = a/(1 − r).

S

=

4/(1 − 1/3)

substitute

S

=

4/(2/3) = 6

simplify

Answer: 6.

C3. Find the geometric mean of 3 and 27.

▶ Reveal full working

Take the square root of the product.

GM

=

√(3 × 27) = √81

GM formula

GM

=

9

simplify

Answer: 9.

C4. Find the sum of the AP 7 + 10 + 13 + … to 15 terms.

▶ Reveal full working

Here a = 7, d = 3, n = 15.

S₁₅

=

(15/2)[2(7) + 14(3)]

substitute

S₁₅

=

(15/2)[14 + 42]

simplify the bracket

S₁₅

=

(15/2)(56) = 420

multiply

Answer: 420.

Practice Set D – HOTS / Word Problems

D1. The sum of the first n terms of an AP is Sₙ = 3n² + 2n. Find the nth term.

▶ Reveal full working

Use aₙ = Sₙ − Sₙ₋₁.

Sₙ

=

3n² + 2n

given

Sₙ₋₁

=

3(n−1)² + 2(n−1)

replace n by n − 1

aₙ

=

Sₙ − Sₙ₋₁

subtract

aₙ

=

6n − 1

simplify

Answer: aₙ = 6n − 1.

D2. Three numbers are in GP. Their product is 216 and their sum is 26. Find them.

▶ Reveal full working

Write the three numbers as a/r, a and ar, so the product is a³.

(a/r)(a)(ar)

=

216

product of the three

a³

=

216, so a = 6

cube root

6(1/r + 1 + r)

=

26

sum of the three

3r² − 10r + 3

=

0

tidy up after substituting

r

=

3 or 1/3

factorise

Answer: The numbers are 2, 6 and 18.

D3. Insert three arithmetic means between 2 and 14.

▶ Reveal full working

With the means, there are 5 terms in AP from 2 to 14.

5 terms: first 2, last 14, so d = (14 − 2)/4 = 3. – find the common difference

The means are 2 + 3, 2 + 6, 2 + 9. – add d each time

That is 5, 8 and 11.

Answer: The three means are 5, 8 and 11.

D4. A ball is dropped from 8 m and each bounce reaches half the previous height. Find the total distance travelled before it comes to rest.

▶ Reveal full working

The downward and upward bounces each form an infinite GP.

Down: 8 + 4 + 2 + … = 8/(1 − ½) = 16 m. – first GP

Up: 4 + 2 + 1 + … = 4/(1 − ½) = 8 m. – second GP

total

=

16 + 8

add the two

total

=

24 m

Answer: The total distance is 24 m.

Chapter Summary

Everything in One Glance

Sequence and Series

A sequence is an ordered list; a series is its sum. The nth term is written aₙ.

Arithmetic Progression

Add d each time. aₙ = a + (n − 1)d and Sₙ = (n/2)[2a + (n − 1)d].

Geometric Progression

Multiply by r each time. aₙ = a rⁿ₋¹ and Sₙ = a(rⁿ − 1)/(r − 1).

Infinite GP

If −1 < r < 1, the infinite sum is finite: S = a/(1 − r).

Means

AM of a, b is (a + b)/2; GM of positive a, b is √(ab).

AM and GM

For positive numbers AM ≥ GM, with equality only when the numbers are equal.

Are You Exam-Ready?

8-Point Exam Quick-Check

Find the 15th term of the AP 2, 5, 8, …

Find the sum of the first 12 terms of the AP 1, 3, 5, …

Find the 5th term of the GP 5, 10, 20, …

Find the sum of the infinite GP 9 + 3 + 1 + …

Find the arithmetic mean of 7 and 19.

Find the geometric mean of 2 and 50.

Which term of the AP 4, 7, 10, … is 49?

State the relationship between the AM and the GM of two positive numbers.

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Class 11 Mathematics Chapter 8: Sequences and Series, Complete Notes and Practice

These free Class 11 Maths Chapter 8 Sequences and Series notes follow the latest NCERT 2026-27 syllabus and give complete, exam-ready coverage for board exams and competitive foundations. Includes worked examples, step-by-step proofs and graded practice, free on SchoolRevise.com.