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Predicting What Comes Next: Exploring Sequences and Progressions – Class 9 Mathematics notes, important questions and MCQs

NCERT Class 9 Mathematics, chapter 8. Notes, key terms, important questions with answers and MCQs.

Chapter overview

A sequence is an ordered list of numbers that follows a rule. Arithmetic progressions add a fixed number each time; geometric progressions multiply by a fixed number.

Sequences and patterns

Find the rule from differences or ratios of consecutive terms.
Square numbers 1, 4, 9, 16…; triangular numbers 1, 3, 6, 10…; Fibonacci 1, 1, 2, 3, 5, 8….
The nth term formula lets you find any term directly.

Arithmetic progression (AP)

Common difference d = a₂ − a₁.
nth term aₙ = a + (n − 1)d.
Sum Sₙ = n/2 [2a + (n − 1)d].

Geometric progression (GP)

Common ratio r = a₂ / a₁.
nth term aₙ = a rⁿ⁻¹.
Example: 3, 6, 12, 24… has r = 2.

Key terms

Sequence: an ordered list of numbers
Common difference: fixed amount added in an AP
Common ratio: fixed number multiplied in a GP

Formulas

AP: aₙ = a + (n − 1)d
GP: aₙ = a rⁿ⁻¹
Sum of first n natural numbers = n(n + 1)/2

Important questions with answers

Find the next two terms: 3, 7, 11, 15, …2 marks

The common difference is 4, so the next terms are 19 and 23.

Is 2, 6, 18, 54, … an AP or a GP? Find its 6th term.3 marks

Each term is 3 times the previous one, so it is a GP with a = 2, r = 3. 6th term = 2 × 3⁵ = 486.

A person saves ₹100 in the first month, ₹150 in the second, ₹200 in the third and so on. How much does the person save in the 12th month and in the whole year?5 marks

Savings form an AP with a = 100, d = 50. 12th month: a₁₂ = 100 + 11 × 50 = ₹650. Total for 12 months: S₁₂ = 12/2 × (100 + 650) = 6 × 750 = ₹4500.

Quick revision

AP adds d; GP multiplies by r.
aₙ = a + (n − 1)d.
Square numbers: n².

Exam tips

Check both differences and ratios when the rule is not obvious.
In Fibonacci, each term = sum of the two before it.

Practice MCQs

Next term of 5, 9, 13, 17, … is
1. 19
2. 20
3. 21
4. 22
Show answer
Answer: (c) 21. Add 4.
Next term of 2, 6, 18, 54, … is
1. 162
2. 108
3. 72
4. 58
Show answer
Answer: (a) 162. Multiply by 3.
Next Fibonacci number after 5, 8, 13 is
1. 18
2. 20
3. 21
4. 26
Show answer
Answer: (c) 21. 8 + 13.
The 12th term of 4, 7, 10, … is
1. 40
2. 37
3. 34
4. 36
Show answer
Answer: (b) 37. 4 + 11 × 3 = 37.
The 6th term of 1, 2, 4, 8, … is
1. 32
2. 16
3. 64
4. 12
Show answer
Answer: (a) 32. 2⁵ = 32.
The 10th triangular number is
1. 45
2. 50
3. 55
4. 100
Show answer
Answer: (c) 55. 10 × 11 / 2 = 55.
Sum of the first 10 terms of 1, 3, 5, … is
1. 110
2. 55
3. 90
4. 100
Show answer
Answer: (d) 100. Sum of first n odd numbers = n².
In an AP, a₃ = 12 and a₇ = 24. Then d is
1. 2
2. 3
3. 4
4. 6
Show answer
Answer: (b) 3. 4d = 12 ⇒ d = 3.

All chapters of Class 9 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.