CBSE Class 9 Maths Revision Notes Chapter 8 Predicting What Comes Next: Exploring Sequences and Progressions

Sequences are ordered lists whose terms may follow a pattern, formula or recursive rule.
Arithmetic and geometric progressions help predict terms using a common difference or common ratio.

Patterns are present in number arrangements, designs, finance, nature and many daily activities. Mathematics uses sequences to describe these ordered patterns and predict later terms.

These CBSE Class 9 Maths Revision Notes Chapter 8 explain explicit rules, recursive rules, arithmetic progressions and geometric progressions. The notes follow the current 2026–27 textbook.

Key Takeaways

• Sequence: An ordered list in which every number is called a term.
• Arithmetic progression: Consecutive terms differ by a fixed number.
• Geometric progression: Consecutive terms have a fixed ratio.
• Natural-number sum: 1 + 2 + ... + n = n(n + 1)/2.

Access Class 9 Maths Chapter 8 Notes in 30 Minutes

Use this order for quick revision:

First 10 minutes: Sequences, notation, explicit and recursive rules
Next 10 minutes: Arithmetic progressions and sums of natural numbers
Final 10 minutes: Geometric progressions, graphs and fractal applications

Ready to practise sequences, APs and GPs with guided solutions?
Access chapter-wise notes, interactive questions and doubt-solving support on the Extramarks Learning App. Sign Up Free

Sequences in Class 9 Maths Chapter 8 Notes

A sequence is an ordered list of numbers or objects. Each member of the list is called a term.

Examples of Sequences

• Natural numbers: 1, 2, 3, 4, 5, ...
• Odd numbers: 1, 3, 5, 7, 9, ...
• Triangular numbers: 1, 3, 6, 10, 15, ...
• Square numbers: 1, 4, 9, 16, 25, ...

The three dots show that the pattern continues.

Terms of a Sequence

The terms are written as:

t₁, t₂, t₃, ..., tₙ

Here:

• t₁ is the first term.
• t₂ is the second term.
• tₙ is the nth term.

For the odd-number sequence:

t₁ = 1

t₂ = 3

t₃ = 5

t₄ = 7

Finite and Infinite Sequences

A finite sequence has a fixed number of terms.

Example:

6, 12, 24, 48, 96

It contains five terms.

An infinite sequence continues without ending.

Example:

1, 2, 3, 4, ...

Increasing and Decreasing Sequences

An increasing sequence has terms that generally become larger.

Example:

2, 5, 8, 11, ...

A decreasing sequence has terms that generally become smaller.

Example:

1, 1/2, 1/3, 1/4, ...

A sequence may also contain negative numbers.

Example:

−7, −3, 1, 5, 9, ...

Common Number Patterns

Recognising common patterns helps in writing formulas.

Odd Numbers

The odd-number sequence is:

1, 3, 5, 7, 9, ...

Its nth term is:

tₙ = 2n − 1

Square Numbers

The square-number sequence is:

1, 4, 9, 16, 25, ...

Its nth term is:

tₙ = n²

Each square number can also be written as a sum of consecutive odd numbers.

For example:

16 = 1 + 3 + 5 + 7

Triangular Numbers

The triangular-number sequence is:

1, 3, 6, 10, 15, ...

Each term is the sum of natural numbers up to that position.

For example:

T₅ = 1 + 2 + 3 + 4 + 5 = 15

Its formula is:

Tₙ = n(n + 1)/2

Explicit Rules in CBSE Class 9 Maths Chapter 8 Notes

An explicit rule gives the nth term directly using the position n.

It allows us to find any term without calculating earlier terms.

Example 1

For:

tₙ = 2n − 1

find the fifth term.

t₅ = 2(5) − 1

= 10 − 1

= 9

Finding a Distant Term

If:

sₙ = 5n − 2

then:

s₁₀₀ = 5(100) − 2

= 500 − 2

= 498

Checking Whether a Number Is a Term

To check whether a number belongs to a sequence:

1. Set the formula equal to the given number.
2. Solve for n.
3. Check whether n is a natural number.

Example 2

Check whether 308 is a term of:

sₙ = 5n − 2

Set:

5n − 2 = 308

5n = 310

n = 62

Since 62 is a natural number, 308 is the 62nd term.

Example 3

Check whether 471 is a term of the same sequence.

5n − 2 = 471

5n = 473

n = 94.6

Since n is not a natural number, 471 is not a term.

Recursive Rules in Class 9 Maths Chapter 8 Notes

A recursive rule defines a term using one or more earlier terms.

The first term or initial terms must be given.

Example 4

The sequence:

1, 4, 7, 10, 13, ...

can be written recursively as:

t₁ = 1

tₙ = tₙ₋₁ + 3, for n ≥ 2

Each new term is obtained by adding 3.

Example 5

Given:

u₁ = 1

uₙ = 2uₙ₋₁ + 3

Find the first four terms.

u₂ = 2(1) + 3 = 5

u₃ = 2(5) + 3 = 13

u₄ = 2(13) + 3 = 29

The sequence is:

1, 5, 13, 29, ...

Explicit Rule and Recursive Rule

Virahanka-Fibonacci Sequence

The Virahanka-Fibonacci sequence is generated by adding the previous two terms.

If:

V₁ = 1

V₂ = 2

then:

Vₙ = Vₙ₋₁ + Vₙ₋₂

The terms are:

1, 2, 3, 5, 8, 13, 21, 34, ...

For example:

V₅ = V₄ + V₃

= 5 + 3

= 8

This sequence was studied in India by Virahanka in connection with poetic metre. It was later studied by Gopala, Hemachandra and Fibonacci.

Arithmetic Progressions in Class 9 Maths Revision Notes Chapter 8

An arithmetic progression is a sequence in which the difference between consecutive terms remains constant.

This fixed difference is called the common difference.

Examples of APs

1, 5, 9, 13, 17, ...

Common difference:

5 − 1 = 4

9 − 5 = 4

Therefore, d = 4.

Another example:

11, 7, 3, −1, −5, ...

Here:

d = −4

An AP may increase, decrease or remain constant.

General Form of an AP

An arithmetic progression has the form:

a, a + d, a + 2d, a + 3d, ...

where:

• a is the first term.
• d is the common difference.

nth Term of an AP

The nth term is:

tₙ = a + (n − 1)d

Example 6

Find the 10th term of:

3, 8, 13, 18, ...

Here:

a = 3

d = 5

t₁₀ = 3 + (10 − 1)(5)

= 3 + 45

= 48

Finding Which Term Has a Given Value

For an AP:

21, 18, 15, ...

find which term is −81.

Here:

a = 21

d = −3

Use:

tₙ = a + (n − 1)d

−81 = 21 + (n − 1)(−3)

−102 = −3(n − 1)

34 = n − 1

n = 35

Therefore, −81 is the 35th term.

Recursive Rule of an AP

An AP can be written recursively as:

t₁ = a

tₙ = tₙ₋₁ + d

For:

1, 5, 9, 13, ...

the recursive rule is:

t₁ = 1

tₙ = tₙ₋₁ + 4

Graph of an Arithmetic Progression

When ordered pairs (n, tₙ) from an AP are plotted, the points lie on a straight line.

For:

1, 5, 9, 13, 17, ...

the points are:

(1,1), (2,5), (3,9), (4,13), (5,17)

These points lie on a straight line because the nth term is a linear expression in n.

Real-Life Application of an AP

A taxi company charges a fixed fee of ₹200 and ₹40 per kilometre.

The fares after 1 km, 2 km and 3 km are:

240, 280, 320, ...

This is an AP with:

a = 240

d = 40

The fare after n kilometres is:

tₙ = 240 + (n − 1)(40)

= 200 + 40n

For 10 km:

t₁₀ = 200 + 40(10)

= ₹600

Sum of the First n Natural Numbers

The sum is:

1 + 2 + 3 + ... + n

Its formula is:

Sₙ = n(n + 1)/2

Short Derivation

Let:

S = 1 + 2 + 3 + ... + n

Write it in reverse:

S = n + (n − 1) + ... + 1

Add the two rows:

2S = n(n + 1)

Therefore:

S = n(n + 1)/2

Example 7

Find the sum of the first 20 natural numbers.

S₂₀ = 20(21)/2

= 10 × 21

= 210

Sum of Consecutive Natural Numbers

To find:

25 + 26 + ... + 58

use:

S₅₈ − S₂₄

= 58(59)/2 − 24(25)/2

= 1711 − 300

= 1411

Triangular Numbers Class 9

Triangular numbers are sums of the first natural numbers.

Their formula is:

Tₙ = n(n + 1)/2

Example 8

Find the 10th triangular number.

T₁₀ = 10(11)/2

= 55

The triangular-number formula and the sum of the first n natural numbers are the same.

Geometric Progressions in Class 9 Chapter 8 Maths Notes

A geometric progression is a sequence in which each term after the first is obtained by multiplying the previous term by a fixed number.

This fixed multiplier is called the common ratio.

Examples of GPs

3, 6, 12, 24, 48, ...

The ratio is:

6/3 = 2

12/6 = 2

Therefore:

r = 2

Another example:

1, −1, 1, −1, ...

Here:

r = −1

General Form of a GP

A geometric progression has the form:

a, ar, ar², ar³, ...

where:

• a is the first term.
• r is the common ratio.

nth Term of a GP

The nth term is:

tₙ = arⁿ⁻¹

Example 9

Find the 10th term of:

5, 25, 125, ...

Here:

a = 5

r = 5

t₁₀ = 5 × 5⁹

= 5¹⁰

Checking Whether a Sequence Is a GP

A sequence is a GP when the ratio of every pair of consecutive terms is constant.

Consider:

5, 15/4, 45/16, 135/64, ...

The ratios are:

(15/4) ÷ 5 = 3/4

(45/16) ÷ (15/4) = 3/4

Therefore, it is a GP with:

a = 5

r = 3/4

Its nth term is:

tₙ = 5(3/4)ⁿ⁻¹

Recursive Rule of a GP

A GP may be written as:

t₁ = a

tₙ = rtₙ₋₁

For:

3, 30, 300, 3000, ...

the recursive rule is:

t₁ = 3

tₙ = 10tₙ₋₁

Graph of a Geometric Progression

Points obtained from a GP generally do not lie on a straight line.

For:

3, 6, 12, 24, 48, ...

the points are:

(1,3), (2,6), (3,12), (4,24), (5,48)

They form a curved pattern because the terms grow by repeated multiplication.

Arithmetic Progression and Geometric Progression

Fractals and Geometric Progressions

A fractal is a pattern that repeats itself at different scales.

The Sierpiński triangle is one example.

Number of Black Triangles

At stages 0, 1, 2, 3, the number of black triangles is:

1, 3, 9, 27, ...

This is a GP with:

a = 1

r = 3

At stage n, the number is:

tₙ = 3ⁿ

Area of the Black Region

If the area at stage 0 is 1, then the areas are:

1, 3/4, (3/4)², (3/4)³, ...

This is a GP with common ratio:

r = 3/4

At stage n, the area is:

sₙ = (3/4)ⁿ

The number of triangles increases, but the total shaded area decreases.

Bouncing Ball as a GP

A ball is dropped from a height of 24 feet. After each bounce, it rises to 3/4 of its previous height.

The heights are:

18, 13.5, 10.125, ...

This is a GP with:

a = 18

r = 3/4

The height after the nth bounce is:

tₙ = 18(3/4)ⁿ⁻¹

Important Formulas in Class 9 Maths Chapter 8

Common Mistakes in Sequences and Progressions

Mistake 1: Confusing Term Number and Term Value

In t₅ = 17, the number 5 is the position and 17 is the value.

Mistake 2: Forgetting n − 1

The formulas are:

tₙ = a + (n − 1)d

and:

tₙ = arⁿ⁻¹

The exponent or multiplier count starts from zero for the first term.

Mistake 3: Using Difference for a GP

An AP has a constant difference. A GP has a constant ratio.

Mistake 4: Accepting a Non-Natural Position

If solving for n gives a fraction or decimal, the number is not a term of the sequence.

Mistake 5: Treating Every Sequence as an AP or GP

Some sequences, such as prime numbers, do not have a constant difference or ratio.

Quick Revision of Predicting What Comes Next

• A sequence is an ordered list.
• A sequence may be finite or infinite.
• An explicit rule finds a term directly.
• A recursive rule uses earlier terms.
• The Virahanka-Fibonacci sequence uses the previous two terms.
• An AP has a constant difference.
• The nth AP term is a + (n − 1)d.
• AP points lie on a straight line.
• The sum of the first n natural numbers is n(n + 1)/2.
• Triangular numbers use the same formula.
• A GP has a constant ratio.
• The nth GP term is arⁿ⁻¹.
• GP points generally do not lie on a straight line.
• Fractal patterns can generate geometric progressions.

Useful Links for Class 9 Maths