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CBSE · Class 9 · Mathematics

Chapter 8: Sequences and Progressions

6 concepts · 48 practice questions with answers · Free to read

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Sequences and number patterns

From daily life

Roll numbers in a class, bus stops on a route and the days of a week are all lists in a fixed order. Many lists follow a pattern.

Simple definition

A sequence is a list of numbers in a definite order. Each number is a term: a₁ first term, a₂ second, aₙ the n-th. A sequence can end (finite) or go on for ever (infinite).

Formula: a₁, a₂, a₃, …, aₙ, …

3›6›9›12›15pattern: add 3 each time

In the figure: 3, 6, 9, 12, 15, … adds 3 each time, so the next term is 18.

Rule 1: Spot the pattern — Look at differences or ratios of neighbouring terms.

  • Easy: 2, 4, 6, 8, …: next 10.
  • Medium: 1, 4, 9, 16, …: squares, next 25.
  • Exam-type: 1, 1, 2, 3, 5, 8, …: add the previous two; next 13.

Rule 2: Terms and positions — Position counts from 1. The term in position n is aₙ.

  • Easy: In 5, 10, 15, 20: a₃ = 15.
  • Medium: Which position holds 40 in 4, 8, 12, …? 10th.
  • Exam-type: Even numbers up to 20 form a sequence of 10 terms.
Remember

Order matters: 1, 2, 3 and 3, 2, 1 are different sequences.

Common mistake

Starting the count of positions from 0.

Memory clue

"List with a rule: sequence."

Practice questions

Easy
  1. 1In the sequence 5, 8, 11, 14, 17, 20, … which term number is 14?
  2. 2In the sequence 6, 10, 14, 18, 22, 26, … which term number is 26?
  3. 3Find the missing term in the sequence 4, 7, 10, 13, □, 19.
Medium
  1. 4Find the missing term in the sequence 16, 24, 32, □, 48, 56.
  2. 5The triangular numbers are 1, 3, 6, 10, 15, … What is the 9th triangular number?
  3. 6The triangular numbers are 1, 3, 6, 10, 15, … What is the 7th triangular number?
Challenging
  1. 7In the sequence of squares 1, 4, 9, 16, …, which term number is 144?
  2. 8In the sequence of squares 1, 4, 9, 16, …, which term number is 169?
Show answers (try them yourself first)
  1. 14
  2. 26
  3. 316
  4. 440
  5. 545
  6. 628
  7. 712
  8. 813

Explicit rule for a sequence

From daily life

A taxi fare table gives the fare for any distance directly from a formula, without calculating the earlier fares.

Simple definition

An explicit rule gives aₙ directly in terms of n. Put n = 1, 2, 3, … to get the terms, or any single term at once.

Formula: aₙ = rule in n, e.g. aₙ = 3n + 1

47101316n=1n=2n=3n=4n=5

In the figure: For aₙ = 3n + 1: n = 1 gives 4, n = 2 gives 7, n = 3 gives 10.

Rule 1: Find terms — Substitute the position n into the rule.

  • Easy: aₙ = 2n − 1: a₅ = 9.
  • Medium: aₙ = n²: a₆ = 36.
  • Exam-type: aₙ = n(n + 1): a₄ = 20.

Rule 2: Find the position — Set the rule equal to the term and solve for n.

  • Easy: aₙ = 3n + 1 = 31 ⇒ n = 10.
  • Medium: aₙ = n² = 49 ⇒ n = 7.
  • Exam-type: Is 50 a term of 2n + 1? 2n = 49, no whole n, so no.
Remember

n must be a positive whole number; a fractional n means the number is not in the sequence.

Common mistake

Mixing up n (position) and aₙ (the term itself).

Memory clue

"Give n, get the term."

Practice questions

Easy
  1. 1A sequence has the rule a(n) = 8n + 7. Find a(13).
  2. 2A sequence has the rule a(n) = 5n + 5. Find a(11).
  3. 3A sequence has the rule a(n) = n² + 4. Find a(7).
Medium
  1. 4For the rule a(n) = 7n, which term number equals 77?
  2. 5For the rule a(n) = 8n, which term number equals 168?
  3. 6A sequence has the rule a(n) = 2ⁿ − 1. Find a(3).
Challenging
  1. 7A sequence has the rule a(n) = 2ⁿ − 3. Find a(6).
  2. 8A sequence has the rule a(n) = (−1)ⁿ × n. Find a(10).
Show answers (try them yourself first)
  1. 1111
  2. 260
  3. 353
  4. 411
  5. 521
  6. 67
  7. 761
  8. 810

Recursive rule for a sequence

From daily life

A bank pays interest on last year's balance. Each year's amount comes from the year before; to know year 5 you need year 4.

Simple definition

A recursive rule gives the first term and says how each term comes from the previous one, such as aₙ = aₙ₋₁ + 3.

Formula: a₁ given; aₙ = f(aₙ₋₁)

2›5›8›11›14+3+3+3+3

In the figure: a₁ = 2 and aₙ = aₙ₋₁ + 3 builds 2, 5, 8, 11, 14, …

Rule 1: Build the terms — Start at a₁ and apply the rule again and again.

  • Easy: a₁ = 5, aₙ = aₙ₋₁ + 2: 5, 7, 9, 11.
  • Medium: a₁ = 3, aₙ = 2aₙ₋₁: 3, 6, 12, 24.
  • Exam-type: a₁ = 1, a₂ = 1, aₙ = aₙ₋₁ + aₙ₋₂: 1, 1, 2, 3, 5.

Rule 2: Write the rule — Find what is added or multiplied each time.

  • Easy: 10, 7, 4, 1: a₁ = 10, aₙ = aₙ₋₁ − 3.
  • Medium: 2, 6, 18, 54: a₁ = 2, aₙ = 3aₙ₋₁.
  • Exam-type: Without a₁ the rule gives many different sequences.
Remember

To get the 20th term from a recursive rule you must compute all earlier terms, unlike an explicit rule.

Common mistake

Forgetting to state the first term.

Memory clue

"First term, then step by step."

Practice questions

Easy
  1. 1A sequence has a(1) = 15 and a(n+1) = a(n) + (−4). Find a(5).
  2. 2A sequence has a(1) = 2 and a(n+1) = a(n) + (−2). Find a(4).
  3. 3A sequence has a(1) = 3 and a(n+1) = 2 × a(n). Find a(5).
Medium
  1. 4A sequence has a(1) = 1, a(2) = 3 and a(n+2) = a(n+1) + a(n). Find a(6).
  2. 5A sequence is defined by a(1) = 8 and a(n+1) = a(n) + 4. Find a(20) without listing all terms.
  3. 6A sequence is defined by a(1) = 4 and a(n+1) = a(n) + 4. Find a(19) without listing all terms.
Challenging
  1. 7A sequence has a(1) = 3 and a(n+1) = 2·a(n) − 1. Find a(5).
  2. 8A sequence has a(1) = 5 and a(n+1) = a(n) + n. Find a(7).
Show answers (try them yourself first)
  1. 1−1
  2. 2−4
  3. 348
  4. 418
  5. 584
  6. 676
  7. 733
  8. 826

Arithmetic progressions

From daily life

Seats in a stadium rise by the same number in each row; a recurring deposit grows by the same amount each month.

Simple definition

An arithmetic progression (AP) has a constant common difference d = aₙ₊₁ − aₙ. With first term a: aₙ = a + (n − 1)d.

Formula: aₙ = a + (n − 1)d

5›9›13›17›21+4+4+4+4a = 5, d = 4

In the figure: In 5, 9, 13, 17, 21 each step adds d = 4. The 10th term is 5 + 9 × 4 = 41.

Rule 1: Is it an AP? — Check that all differences are equal.

  • Easy: 3, 7, 11, 15: d = 4, yes.
  • Medium: 1, 2, 4, 8: differences 1, 2, 4, no.
  • Exam-type: 10, 6, 2, −2: d = −4, yes (decreasing).

Rule 2: Find a term — Use aₙ = a + (n − 1)d.

  • Easy: a = 2, d = 5: a₁₀ = 47.
  • Medium: a = 50, d = −3: a₈ = 29.
  • Exam-type: Which term of 4, 9, 14, … is 94? 4 + (n − 1)5 = 94 gives n = 19.
Remember

In an AP, each term is the average of its two neighbours.

Common mistake

Using n instead of (n − 1) in the formula.

Memory clue

"First plus (n minus 1) steps."

Practice questions

Easy
  1. 1Find the 18th term of the AP with first term 1 and common difference 7.
  2. 2The first term of an AP is 25 and its common difference is 5. Find its 28th term.
  3. 3How many terms are there in the AP 11, 17, 23, …, 107?
Medium
  1. 4How many terms are there in the AP 7, 13, 19, …, 103?
  2. 5In an AP the 3rd term is 11 and the 7th term is 23. Find the 11th term.
  3. 6The 5th term of an AP is 26 and the common difference is 4. Find the first term.
Challenging
  1. 7Three numbers are in AP and their sum is 36. What is the middle number?
  2. 8For what value of x are the numbers x, 2x + 2 and 4x − 2 in AP?
Show answers (try them yourself first)
  1. 1120
  2. 2160
  3. 317
  4. 417
  5. 535
  6. 610
  7. 712
  8. 86
Live figure

Arithmetic progressions

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Sum of natural numbers and of AP terms

From daily life

A boy adds 1 + 2 + … + 100 quickly by pairing 1 with 100, 2 with 99, and so on: 50 pairs of 101.

Simple definition

Sum of first n natural numbers: n(n + 1)/2. Sum of first n terms of an AP: Sₙ = n/2 × [2a + (n − 1)d] or n/2 × (first + last).

Formula: Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)

1234561+6 = 2+5 = 3+4 = 73 pairs: 21

In the figure: Pairs of equal sum 7 make 1 + 2 + … + 6 = 3 × 7 = 21.

Rule 1: Natural numbers — Use n(n + 1)/2, or subtract two such sums for a range.

  • Easy: 1 + 2 + … + 100 = 5050.
  • Medium: 1 + 2 + … + 20 = 210.
  • Exam-type: Sum of first 10 odd numbers = 10² = 100.

Rule 2: AP sums — Use the first-plus-last form when the last term is known.

  • Easy: AP 2, 5, 8, … 10 terms: S = 5 × [4 + 27] = 155.
  • Medium: 3 + 7 + … + 39 (10 terms): 5 × 42 = 210.
  • Exam-type: a = 1, d = 1, n = 50: 1275.
Remember

Check: S₁ = a, and Sₙ − Sₙ₋₁ = aₙ.

Common mistake

Forgetting the ½ in the formula.

Memory clue

"Average of first and last, times the count."

Practice questions

Easy
  1. 1Find the sum 1 + 2 + 3 + … + 80.
  2. 2Find the sum 1 + 2 + 3 + … + 30.
  3. 3Find the sum of the first 25 odd natural numbers (1 + 3 + 5 + …).
Medium
  1. 4Find the sum of all multiples of 2 that are less than or equal to 120.
  2. 5Find the sum of all multiples of 7 that are less than or equal to 50.
  3. 6How many natural numbers, starting from 1, must be added to get a sum of 351?
Challenging
  1. 7How many natural numbers, starting from 1, must be added to get a sum of 45?
  2. 8The sum of the first 14 terms of an AP is 329 and its first term is 4. Find the common difference.
Show answers (try them yourself first)
  1. 13240
  2. 2465
  3. 3625
  4. 43660
  5. 5196
  6. 626
  7. 79
  8. 83

Geometric progressions

From daily life

A rumour told to 2 people, each telling 2 more, spreads 1, 2, 4, 8, 16, … Each step multiplies by the same number.

Simple definition

A geometric progression (GP) has a constant common ratio r = aₙ₊₁ ÷ aₙ. The n-th term is aₙ = a · rⁿ⁻¹. Growth like this is called exponential.

Formula: aₙ = a · rⁿ⁻¹

2›4›8›16›32×2×2×2×2a = 2, r = 2

In the figure: In 2, 4, 8, 16, 32 each step multiplies by r = 2. The 6th term is 2 × 2⁵ = 64.

Rule 1: Find r and terms — Divide a term by the one before it.

  • Easy: 3, 6, 12, 24: r = 2, next 48.
  • Medium: 81, 27, 9, 3: r = 1/3, next 1.
  • Exam-type: a = 5, r = 2: a₄ = 5 × 8 = 40.

Rule 2: AP or GP? — Equal differences: AP. Equal ratios: GP.

  • Easy: 4, 7, 10, 13: AP (d = 3).
  • Medium: 4, 12, 36, 108: GP (r = 3).
  • Exam-type: 2, 4, 7, 11: neither.
Remember

A GP grows (or shrinks) much faster than an AP.

Common mistake

Using aₙ = a · r · n or a · rⁿ instead of a · rⁿ⁻¹.

Memory clue

"AP adds, GP multiplies."

Practice questions

Easy
  1. 1A GP has first term 5 and common ratio 3. Find its 5th term.
  2. 2Is the list 3, 6, 12, 24, 48 a geometric progression?
  3. 3A GP has first term 2 and common ratio 3. Find the 8th term.
Medium
  1. 4In a GP the 2nd term is 6 and the 4th term is 54. Find the common ratio (positive).
  2. 5A culture has 6 bacteria at noon, and the number doubles every hour. How many bacteria are there 7 hours later?
  3. 6A culture has 2 bacteria at noon, and the number doubles every hour. How many bacteria are there 3 hours later?
Challenging
  1. 7Find the sum of the first 5 terms of the GP with first term 2 and common ratio 2.
  2. 8A GP has first term 3 and common ratio 2. Which is the first term number whose value is greater than 200?
Show answers (try them yourself first)
  1. 1405
  2. 2Yes, it is a GP
  3. 34374
  4. 43
  5. 5768
  6. 616
  7. 762
  8. 88