Practise Class 9 Maths Part 1 Chapter 8 – Predicting What Comes Next: Exploring Sequences and Progressions with this Ganita Manjari Chapter 8 Worksheet. This worksheet includes important questions and revision exercises to strengthen understanding of sequences and progressions, improve mathematical practice and support effective preparation for class tests and examinations.

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MATHEMATICS WORKSHEET

Chapter 8: Predicting What Comes Next

Exploring Sequences and Progressions

Name: __________________________

Class: __________

Date: __________

Instructions:

Answer all questions.
Show your working wherever required.
Use the appropriate formula for AP and GP questions.

A. Sequences and Explicit Rules

1. Can you describe the pattern in each sequence and predict the next few numbers?

(i) 1, 2, 3, 4, 5, 6, …
(ii) 1, 3, 5, 7, 9, 11, …
(iii) 1, 3, 6, 10, 15, 21, …
(iv) 1, 4, 9, 16, 25, 36, …

2. Consider the sequence 1, 4, 7, 10, 13, …
(a) Find the next four terms.
(b) Find the first 10 terms of the sequence obtained by adding all the terms up to a given term.

3. Write t₅, t₆, t₇ and t₈ for the sequence of triangular numbers.

4. Can you find the rule describing the nth term of the sequence of square numbers?

5. Using the explicit rule uₙ = 2n − 1, find:
(a) the 53rd term
(b) the 108th term
(c) the 1170th term

6. Consider the expression tₙ = 3n − 7.
(i) Find its first, second, third, 12th, 18th and 50th terms.
(ii) Which term of the sequence is 332?
(iii) Is 557 a term of this sequence? Why or why not?

B. Recursive Rules

7. Write the next two terms of the sequence:

1, 2, 3, 5, 8, 13, 21, 34, …

8. Write a recursive rule for the sequence:

2, 5, 8, 11, …

9. Write a recursive rule for the sequence:

−5, −1, 3, 7, …

C. Exercise Set 8.1

10. Find the first five terms of the sequence in which the nth term is given by:
(i) tₙ = 3n − 4
(ii) tₙ = 2 − 5n
(iii) tₙ = n² − 2n + 3 for n ≥ 1.

11. Find the 10th and 15th terms of the sequence tₙ = 5n − 3 for n ≥ 1.

12. Determine whether 97 and 172 are terms of the sequence tₙ = 5n − 3 for n ≥ 1.

13. Which term of the sequence tₙ = 5n − 3 for n ≥ 1 is 607?

14. A sequence is given by the recursive rule t₁ = −5, tₙ₊₁ = tₙ + 3 for n ≥ 1.
Find the first five terms. Is 52 a term of this sequence? If so, which term is it?

15. Let T₁ = 1, T₂ = 2, T₃ = 4, and Tₙ = Tₙ₋₁ + Tₙ₋₂ + Tₙ₋₃ for n ≥ 4.
Find T₄, T₅, T₆, T₇ and T₈.

D. Arithmetic Progressions

16. Verify that the following sequences are arithmetic progressions and write their nth terms. What do you observe when you plot the ordered pairs emerging from them?
(i) 2, 5, 8, 11, …
(ii) −5, −1, 3, 7, …

17. Using the formula tₙ = a + (n − 1)d, find the nth term of:
(i) 1/2, 5/2, 9/2, 13/2, …
(ii) 1.5, 3.5, 5.5, 7.5, …

18. Find recursive rules for the APs in the previous question.

E. Sum of Natural Numbers

19. Use the formula for the sum of the first n natural numbers to find S₂₀, S₅₀ and S₁₀₀₀.

20. Find the sum:

25 + 26 + 27 + … + 58

21. Find the 10th, 17th and 80th triangular numbers.

F. Exercise Set 8.2

22. Find the 10th and 26th terms of the AP: 3, 8, 13, 18, ….

23. Which term of the AP: 21, 18, 15, … is −81? Also, is 0 a term of this AP? Give reasons for your answer.

24. Find the nth term of the AP: 11, 8, 5, 2, … Write the recursive rule for this AP.

25. An AP consists of 50 terms in which the 3rd term is 12 and the last term is 106. Find the 29th term.

26. How many 2-digit numbers are divisible by 3? What is the sum of all these 2-digit numbers?

27. Harish started work at an annual salary of ₹5,00,000 and received an increment of ₹20,000 each year. After how many years did his income reach ₹7,00,000?

G. Geometric Progressions

28. Check whether the following sequences are geometric progressions and find their nth terms.
(i) 2, 10, 50, 250, …
(ii) 4, 8/3, 16/9, 32/27, …
(iii) 3, −3/2, 3/4, −3/8, …

29. Can you find a recursive rule for the formula tₙ = 3 × 10ⁿ⁻¹ that generates the geometric progression 3, 30, 300, 3000, …?

H. Exercise Set 8.3

30. Find the 12th term of a GP with common ratio 2, whose 8th term is 192.

31. Find the 10th and nth terms of the GP: 5, 25, 125, ….

32. A sequence is given by the recursive rule t₁ = 2, tₙ₊₁ = 3tₙ − 2 for n ≥ 1. Which term of the sequence is 730?

33. Which term of the GP: 2, 6, 18, … is 4374? Write the explicit formula as well as the recursive formula for the nth term.

34. A ball is dropped from a height of 80 metres. After hitting the ground, it bounces back to 60% of the height from which it fell. It continues bouncing in this way — each time rising to 60% of the previous height.

(i) What height does the ball reach after the 5th bounce?
(ii) What is the total vertical distance the ball has travelled by the time it hits the ground for the 6th time?

35. Which term of the sequence 2, 2√2, 4, … is 128?

I. End-of-Chapter Practice

36. Find the 31st term of an AP whose 11th term is 38 and 16th term is 73.

37. Determine the AP whose third term is 16 and whose 7th term exceeds the 5th term by 12.

38. How many three-digit numbers are divisible by 7?

39. How many multiples of 4 lie between 10 and 250?

40. Find a GP for which the sum of the first two terms is −4 and the fifth term is 4 times the third term.

41. Find all possible ways of expressing 100 as the sum of consecutive natural numbers.

42. The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of the 2nd hour, 4th hour and nth hour?

43. The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.

44. Find the smallest value of n such that the sum of the first n natural numbers is greater than 1,000.

45. Which term of the GP: 2, 8, 32, … is 131072? Write the explicit formula as well as the recursive formula for the nth term.

46. The sum of the first three terms of a GP is 13/12 and their product is −1. Find the common ratio and the terms.

47. If the 4th, 10th and 16th terms of a GP are x, y and z respectively, prove that x, y, z are in GP.

48. The sum of the first three terms of a geometric progression is 26, and the sum of their squares is 364. Find the terms of the GP.

49. Suppose P₁ = 1, P₂ = 2 and for n > 2, Pₙ = P₁ + P₂ + … + Pₙ₋₁ + 1.Find the values of P₁, P₂, …, P₈. Can you find a simpler recursive formula for Pₙ? Can you give an explicit formula?

50. Suppose W₁ = 1, W₂ = 2 and for n > 2, Wₙ = W₁ + W₂ + … + Wₙ₋₂ + 2.Find the values of W₁, W₂, …, W₈. Do you recognise this sequence?

End of Worksheet
Chapter 8 • Sequences and Progressions

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Class 9 Maths Part 1 Chapter 8 – Predicting What Comes Next: Exploring Sequences and Progressions | Ganita Manjari Worksheet