Question Bank — Predicting What Comes Next: Exploring Sequences and Progressions

NCERT Class 9 Mathematics · Chapter 8 · CBSE Board Pattern

Class 9 Mathematics Chapter 8 Question Bank CBSE Board Pattern

Section A — MCQs (10 questions, 1 mark each)

  1. The sequence given by the explicit rule \( t_n = 3n - 4 \) has its fifth term equal to
    (a) 11 (b) 15 (c) 19 (d) 23

  2. Which of the following is an arithmetic progression?
    (a) 2, 6, 18, 54, …
    (b) 5, 8, 11, 14, …
    (c) 1, 4, 9, 16, …
    (d) 3, 6, 12, 24, …

  3. In the sequence defined by the recursive rule \( u_1 = 1 \), \( u_n = 2u_{n-1} + 3 \) for \( n \geq 2 \), the fourth term is
    (a) 13 (b) 29 (c) 61 (d) 125

  4. The common difference of the AP 11, 7, 3, −1, … is
    (a) −4 (b) 4 (c) −3 (d) 3

  5. For the GP 3, 6, 12, 24, … the common ratio is
    (a) 2 (b) 3 (c) 4 (d) ½

  6. The expression for the nth term of an AP with first term a and common difference d is
    (a) \( a + nd \) (b) \( a + (n-1)d \) (c) \( a + nd - 1 \) (d) \( a + (n+1)d \)

  7. The sum of the first n natural numbers is given by
    (a) \( n(n-1)/2 \) (b) \( n(n+1)/2 \) (c) \( n^2/2 \) (d) \( n(n+2)/2 \)

  8. In the sequence of triangular numbers, the nth term equals
    (a) \( n^2 \) (b) \( n(n+1)/2 \) (c) \( n(n-1)/2 \) (d) \( 2n-1 \)

Assertion-Reason Questions

  1. Assertion (A): The sequence 5, 15/4, 45/16, 135/64, … is a geometric progression.
    Reason (R): The ratio of every pair of consecutive terms is constant and equal to 3/4.
    (a) Both A and R are true and R is the correct explanation of A.
    (b) Both A and R are true but R is not the correct explanation of A.
    (c) A is true but R is false.
    (d) A is false but R is true.

  2. Assertion (A): The explicit formula \( t_n = 5n-3 \) can be used to find any term of the sequence without knowing the previous terms.
    Reason (R): An explicit formula expresses the nth term directly in terms of n.
    (a) Both A and R are true and R is the correct explanation of A.
    (b) Both A and R are true but R is not the correct explanation of A.
    (c) A is true but R is false.
    (d) A is false but R is true.

Section B — Very Short Answer (6 questions, 2 marks each)

  1. Write the first five terms of the sequence whose nth term is given by \( t_n = 2-5n \).

  2. Using the explicit rule \( u_n = 2n-1 \), find the 53rd term of the sequence of odd numbers.

  3. Determine whether 97 is a term of the sequence \( t_n = 5n-3 \). Give reason.

  4. Find the first four terms of the sequence given by the recursive rule \( t_1 = -5 \), \( t_{n+1} = t_n + 3 \) for \( n \geq 1 \).

  5. Write the explicit formula for the nth term of the AP ½, 5/2, 9/2, 13/2, … .

  6. State the recursive rule for the AP whose explicit formula is \( t_n = a + (n-1)d \).

Section C — Short Answer (5 questions, 3 marks each)

  1. Find the 10th and 15th terms of the sequence \( t_n = 5n-3 \).

  2. Which term of the AP 21, 18, 15, … is −81? Is 0 a term of this AP? Justify.

  3. The first term of an AP is 3 and the common difference is 5. Find the 26th term and also write the recursive rule.

  4. Using the formula for the sum of the first n natural numbers, find the sum of the first 20 natural numbers. Also verify by adding the first 10 and the next 10 separately.

  5. Check whether the sequence 2, 10, 50, 250, … is a GP and write its nth term.

Section D — Long Answer (3 questions, 5 marks each)

  1. An AP consists of 50 terms. The 3rd term is 12 and the last term is 106.
    (i) Find the first term and the common difference.
    (ii) Find the 29th term.
    (iii) Write both the explicit and recursive formulae for this AP.

  2. A GP is given by the recursive rule \( t_1 = 2 \), \( t_{n+1} = 3t_n - 2 \) for \( n \geq 1 \).
    (i) Find the first five terms.
    (ii) Which term of the sequence is 730?
    (iii) Write the explicit formula for the nth term.

  3. A ball is dropped from a height of 80 m. After each bounce it rises to 60 % of the previous height.
    (i) Write the sequence of heights after the first five bounces.
    (ii) Show that the sequence is a GP and find its common ratio.
    (iii) Find the height after the 5th bounce using the explicit formula.

Section E — Case/Source-Based (2 questions, 4 marks each)

Case 1

A taxi company charges a fixed booking fee of ₹200 plus ₹40 per kilometre travelled.

(i) Write the sequence of total fares after travelling 1 km, 2 km, 3 km and 4 km.
(ii) Show that the sequence is an AP and state the first term and common difference.
(iii) Find the total fare for a 10 km journey using the explicit formula.
(iv) Write the recursive rule for this sequence.

Case 2

A ball is dropped from a height of 24 feet. After each bounce it rises to ¾ of the previous height.

(i) Write the sequence of maximum heights after the first four bounces.
(ii) Show that the sequence forms a GP and state the common ratio.
(iii) Find the height after the 7th bounce.
(iv) After how many bounces does the height first fall below 1/6 of the original height?

▶ Answer Key Attempt all questions first,
then tap to reveal

Section A

  1. (a) 11
  2. (b) 5, 8, 11, 14, …
  3. (b) 29
  4. (a) −4
  5. (a) 2
  6. (b) \( a + (n-1)d \)
  7. (b) \( n(n+1)/2 \)
  8. (b) \( n(n+1)/2 \)
  9. (a) Both true and R explains A
  10. (a) Both true and R explains A

Section B

  1. −3, −8, −13, −18, −23 (correct first term 1 mark, next four terms 1 mark)
  2. 105 (substitution of n = 53 — 1 mark, correct value — 1 mark)
  3. No; solving 5n − 3 = 97 gives n = 20 (natural number) but 97 does not satisfy (correct equation 1 mark, conclusion 1 mark)
  4. −5, −2, 1, 4 (each correct term ½ mark)
  5. \( t_n = \frac12 + (n-1)\times 2 \) or \( t_n = 2n-3/2 \) (correct a and d 1 mark, formula 1 mark)
  6. \( t_1 = a \), \( t_n = t_{n-1} + d \) for n ≥ 2 (both parts correct 1 mark each)

Section C

  1. 47 and 72 (10th term 1½ marks, 15th term 1½ marks)
  2. 36th term; 0 is not a term (equation solving 2 marks, conclusion 1 mark)
  3. 128; recursive rule \( t_1 = 3 \), \( t_n = t_{n-1} + 5 \) (explicit term 2 marks, recursive 1 mark)
  4. 210 (formula application 2 marks, verification 1 mark)
  5. Yes, GP with \( t_n = 2 \times 5^{n-1} \) (identification 1 mark, formula 2 marks)

Section D

  1. a = 2, d = 2; 58th term = 58; explicit \( t_n = 2 + (n-1)\times 2 \), recursive \( t_1 = 2 \), \( t_n = t_{n-1} + 2 \) (each sub-part carries appropriate partial marks)
  2. 2, 4, 10, 28, 82; 730 is the 7th term; explicit \( t_n = 3^n - 1 \) (step-wise computation and verification 5 marks)
  3. 18, 10.8, 6.48, 3.888, 2.3328 m; r = 0.6; height after 5th bounce = 80 × (0.6)^5 = 6.2208 m (full working shown)

Section E

Case 1

(i) 240, 280, 320, 360
(ii) AP, a = 240, d = 40
(iii) ₹600
(iv) t₁ = 240, tₙ = tₙ₋₁ + 40

Case 2

(i) 18, 13.5, 10.125, 7.59375 ft
(ii) GP, r = 3/4
(iii) 24 × (3/4)^7 ≈ 3.203 ft
(iv) After 7 bounces (detailed substitution shown with marking for each step)

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All questions are answerable from the NCERT chapter text. Reviewed by GFIS faculty.