NCERT Solutions for Class 9 Maths Chapter 8 Predicting What Comes Next Exploring Sequences and Progressions (PDF) – 2026-27

NCERT Solutions for Class 9 Maths Chapter 8 Predicting What Comes Next Exploring Sequences and Progressions provide clear, step-by-step answers to every exercise and in-text question from the chapter Predicting What Comes Next Exploring Sequences and Progressions of the NCERT textbook Ganita Manjari. Prepared by subject experts as per the latest NCERT (CBSE) syllabus for 2026-27, these NCERT Solutions for Class 9 Maths help you understand each concept, write exam-ready answers, and check your own solutions. You can read them online below or download the free Class 9 Maths Chapter 8 question-answer PDF.

NCERT Solutions for Class 9 Maths Chapter 8 Predicting What Comes Next Exploring Sequences and Progressions

• Class: Class 9
• Subject: Maths
• Chapter: Chapter 8 – Predicting What Comes Next Exploring Sequences and Progressions
• Textbook: Ganita Manjari (NCERT)
• Study material: NCERT Solutions – questions with answers, free PDF

These solutions answer all the exercise questions of Chapter 8 Predicting What Comes Next Exploring Sequences and Progressions — including the in-text questions, short-answer and long-answer questions, and activities — with complete explanations so you can follow the method, not just the final answer. Read the full solutions below.

NCERT Solutions Class 9 Maths Chapter 8 Predicting What Comes Next Exploring Sequences and Progressions View Download

NCERT Solutions for Class 9 Maths Chapter 8 PDF Download

You can read the NCERT Solutions for Class 9 Maths Chapter 8 online above, or download the complete question-answer PDF to study Predicting What Comes Next Exploring Sequences and Progressions offline at any time.

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NCERT Solutions for Class 9 Maths Chapter 8 PDF Download Link – Click Here to Download Solutions PDF

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Questions Covered in This Chapter

These NCERT Solutions answer all 62 questions of this chapter. The questions solved are:

1. Can you describe the pattern in each of the above sequences? Can you predict the next few numbers in these sequences?
2. Can you think of other finite sequences that you see in your daily life?
3. [Fig. 8.1] Can you draw the patterns for the next two terms of the sequence?
4. [Fig. 8.2] This interesting relationship between the odd numbers and square numbers can be represented by the diagram in Fig. 8.2. Can you explain the relationship?
5. Exercise: Consider the sequence 1, 4, 7, 10, 13, … Can you predict the next four terms? Can you derive the first 10 terms of the sequence obtained by adding all the terms up to a given term of this sequence? (Hint: The first term is 1. The second term is 1 + 4 = 5, the third term is 1 + 4 + 7 = 12, and so on.)
6. Exercise: Can you write t5, t6, t7 and t8 for the sequence of triangular numbers?
7. Can you think of any other kinds of sequences? List out five different types of sequences and discuss their properties with your friends.
8. Why is it useful to have an explicit formula for the nth term of a sequence?
9. Can you find the rule describing the nth term of the sequence of square numbers?
10. Exercise: Using the explicit rule un = 2n – 1, find the 53rd term, the 108th term, and the 1170th term of the odd number sequence.
11. Here is the sequence of the first ten prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Do you see any pattern in this sequence? Can you think of a rule that can predict the next few prime numbers?
12. Exercise: 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?
13. [Virahānka–Fibonacci sequence 1, 2, 3, 5, 8, 13, 21, 34, …] Can you write the next two terms of this sequence?
14. Find the first five terms of the sequence in which the nth term is given by (i) tn = 3n – 4, (ii) tn = 2 – 5n, and (iii) tn = n² – 2n + 3 for n ≥ 1.
15. Find the 10th and 15th terms of the sequence tn = 5n – 3 for n ≥ 1.
16. Determine whether 97 and 172 are terms of the sequence tn = 5n – 3 for n ≥ 1.
17. Which term of the sequence tn = 5n – 3 for n ≥ 1 is 607?
18. A sequence is given by the recursive rule t1 = – 5, tn+1 = tn + 3 for n ≥ 1. Find the first five terms of the sequence. Is 52 a term of this sequence? If so, which term is it?
19. 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.
20. [Fig. 8.3, growing pattern of squares 1, 5, 9, 13] Can you predict the number of squares in Stages 5 and 6 of the sequence? In Stages 10, 11 and 12? In Stage 20? At any stage?
21. Consider all the sequences we have discussed so far in this chapter. Which ones are arithmetic progressions and which ones are not? Can you justify your claim?
22. Exercise: 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, …
23. Exercise: Using the formula tn = a + (n – 1) × d, find the nth term of the following arithmetic progressions. (i) 1/2, 5/2, 9/2, 13/2, … (ii) 1.5, 3.5, 5.5, 7.5, …
24. Exercise: Find recursive rules for the APs in the previous exercises.
25. Can the same approach be used to find the sum of 1 + 2 + 3 + … + 100?
26. Can you use this formula to find S20, S50 or S1000?
27. Let us revisit the sequence tn of triangular numbers 1, 3, 6, 10, 15, … shown in Fig. 8.1. Note that the nth term of this sequence is the sum of the first n natural numbers. Thus tn = n(n + 1)/2. Can you use this to find the 10th, 17th and 80th triangular numbers?
28. Find the 10th and 26th terms of the AP: 3, 8, 13, 18, ….
29. Which term of the AP : 21, 18, 15, … is – 81? Also, is 0 a term of this AP? Give reasons for your answer.
30. Find the nth term of the AP: 11, 8, 5, 2 … Write the recursive rule for this AP.
31. An AP consists of 50 terms in which the 3rd term is 12 and the last term is 106. Find the 29th term. (Hint: If ‘a’ is the first term and ‘d’ the common difference, then we arrive at the equations a + 2d = 12 and a + 49d = 106. Solve this pair of linear equations for ‘a’ and ‘d’.)
32. How many 2-digit numbers are divisible by 3? What is the sum of all these 2-digit numbers?
33. 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?
34. A child arranges marbles in rows so that the first row has 1 marble, the second has 2 marbles, the third has 3, and so on up to 25 rows. How many marbles does the child use in all?
35. [Fig. 8.6, growing pattern of squares 3, 6, 12, 24] Can you predict the number of squares in Stages 5 and 6 of the pattern? In Stages 10, 11 and 12? In Stage 20? At any stage? How is this different from the growing pattern in Fig. 8.3?
36. Exercise: 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, …
37. Exercise: Can you find a recursive rule for the formula tn = 3 × 10ⁿ⁻¹ that generates the geometric progression 3, 30, 300, 3000, … ?
38. Observe the Sierpiński triangle and try to answer the following questions. (a) How many black triangles are there in Stages 0 to 3 of Fig. 8.7? (b) Can you predict the number of black triangles at Stages 4 and 5? (c) Can you find a rule for the number of black triangles at the nth stage? (d) Suppose the area of the triangle (that is, the black region) in Stage 0 is 1 square unit. What is the area of the black region in Stages 1, 2 and 3? What will be the area of the black region in Stages 4 and 5? Find a rule for the area of the black region at the nth stage. What happens to this area as n, the number of stages, goes on increasing?
39. The number of black triangles increases very quickly as the stage numbers increase. Can you explain why?
40. Can you explain why the area of the black region at Stage n will be (3/4)ⁿ?
41. Find the 12th term of a GP with common ratio 2, whose 8th term is 192.
42. Find the 10th and nth terms of the GP: 5, 25, 125, … .
43. A sequence is given by the recursive rule t1 = 2, tn+1 = 3tn – 2 for n ≥ 1. Which term of the sequence is 730?
44. Which term of the GP: 2, 6, 18, … is 4374? Write the explicit formula as well as the recursive formula for the nth term.
45. 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?
46. Which term of the sequence 2, 2√2, 4, … is 128?
47. Fig. 8.12 shows Stages 0 to 3 of the Sierpiński square carpet. Stage 0 of this fractal is a square sheet of paper. To construct Stage 1, each side of the square is trisected and the points of trisection of opposite sides are joined to obtain nine smaller squares. The centre square is then removed and the 8 smaller squares are retained, leaving a square hole in the centre. The same process is repeated on the eight smaller shaded squares to obtain Stage 2 and so on. Look at Fig. 8.12 and try to answer the following questions. (i) How many red squares are there in Stages 0 to 3? (ii) Can you predict the number of red squares in Stages 4 and 5? (iii) Can you find a rule for the number of red squares at the nth stage? Write the explicit formula as well as the recursive formula for the number of red squares at any stage. (iv) Suppose the area of the square in Stage 0 is 1 square unit. What is the area of the red region in Stages 1, 2 and 3? What will be the area of the red region in Stages 4 and 5? Find the explicit as well as the recursive formula for the area of the red region at the nth stage. What happens to this area as n, the number of stages, goes on increasing?
48. Find the 31st term of an AP whose 11th term is 38 and 16th term is 73.
49. Determine the AP whose third term is 16 and whose 7th term exceeds the 5th term by 12.
50. How many three-digit numbers are divisible by 7? (Hint: All three-digit numbers divisible by 7 form an AP. Find the smallest and largest such three-digit numbers.)
51. How many multiples of 4 lie between 10 and 250? (Hint: All multiples of 4 form an AP. Find the smallest and largest multiples of 4 between 10 and 250.)
52. Find a GP for which the sum of the first two terms is – 4 and the fifth term is 4 times the third term.
53. Find all possible ways of expressing 100 as the sum of consecutive natural numbers.
54. 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?
55. 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.
56. Find the smallest value of n such that the sum of the first n natural numbers is greater than 1,000.
57. Which term of the GP: 2, 8, 32, … is 131072? Write the explicit formula as well as the recursive formula for the nth term.
58. 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.
59. If the 4th, 10th and 16th terms of a GP are x, y and z respectively, prove that x, y, z are in GP.
60. 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.
61. 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?
62. 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?

Chapter at a Glance

• A sequence is an ordered list of numbers; each number is a term . The notation t 1 , t 2 , t 3 , … ties a term to its position, so t 4 = 7 says "the term in the 4th place is 7". Sequences may be finite (6, 12, 24, 48, 96) or infinite (1, 2, 3, 4, …).
• An explicit rule computes t n straight from the position n — t n = 2n – 1 gives the 1170th odd number without listing the first 1169. It also runs backwards: solving t n = k tells you whether k is a term, and where. The answer counts only if n comes out a natural number.
• A recursive rule gives a term from earlier terms, e.g. t 1 = 1, t n = t n–1 + 3. It may reach back further than one step: the Virahānka–Fibonacci sequence uses V n = V n–1 + V n–2 , giving 1, 2, 3, 5, 8, 13, 21, 34, … It was set down by Virahānka in the 7th century CE in the Vṛttajātisamuchaya , studied by Gopāla and Hemachandra, and only later by Fibonacci.
• An arithmetic progression (AP) has a constant common difference d between consecutive terms: a, a + d, a + 2d, …, with t n = a + (n – 1)d and recursive form t 1 = a, t n = t n–1 + d. Because each step adds the same amount, the plotted points (n, t n ) lie on a straight line.
• The sum of the first n natural numbers is S n = n(n + 1)/2. Write the sum forwards and backwards, add the two lines: every column totals n + 1, and there are n columns, so 2S = n(n + 1). The same fact is a 7 × 6 array of circles cut by a staircase. The triangular numbers 1, 3, 6, 10, 15, … are exactly these sums, so t n = n(n + 1)/2. The result appears in Āryabhaṭa's Āryabhaṭīya , Chapter 2, Verse 19.
• A geometric progression (GP) has a constant common ratio r: a, ar, ar 2 , …, with t n = ar n–1 and recursive form t 1 = a, t n = r · t n–1 . Its plotted points curve away from any straight line. Fractals supply GPs of both kinds: in the Sierpiński triangle the count of black triangles is 3 n (growing fast) while the black area is (3/4) n (shrinking towards 0).

How to Download NCERT Solutions for Class 9 Maths Chapter 8 PDF

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NCERT Solutions for Class 9 Maths – All Chapters

There are more chapters to study besides Predicting What Comes Next Exploring Sequences and Progressions in Maths. Here are the NCERT Solutions for all chapters of Class 9 Maths.

• Chapter 1 Orienting Yourself the Use of Coordinates
• Chapter 2 Introduction to Linear Polynomials
• Chapter 3 The World of Numbers
• Chapter 4 Exploring Algebraic Identities
• Chapter 5 I M Up and Down and Round and Round
• Chapter 6 Measuring Space Perimeter and Area
• Chapter 7 The Mathematics of Maybe Introduction to Probability
• Chapter 8 Predicting What Comes Next Exploring Sequences and Progressions

NCERT Solutions for Class 9 – All Subjects

Just like Chapter 8 of Maths, you can get the exercise questions with answers for every other subject of Class 9. Here are the NCERT Solutions for all subjects of Class 9.

• English
• Hindi
• Maths
• Sanskrit
• Science
• Social Science

NCERT Solutions for Class 9 Maths Chapter 8 – An Overview

The key highlights of this study material are as follows.

Aspects	Details
Class	Class 9
Subject	Maths
Chapter Number	Chapter 8
Chapter Name	Predicting What Comes Next Exploring Sequences and Progressions
Book Name	Ganita Manjari
Book By	NCERT (National Council of Educational Research and Training)
Educational Resource Here	NCERT Solutions of Class 9 Maths Chapter 8 for all exercises
More Questions Answers of This Subject	NCERT Solutions for Class 9 Maths
Download Book Chapter	NCERT Book Class 9 Maths
All Questions Answers For This Class	NCERT Solutions for Class 9
Complete Solutions	NCERT Solutions

NCERT Solutions for Class 9 Maths Chapter 8 Predicting What Comes Next Exploring Sequences and Progressions – FAQs

What are the NCERT Solutions for Class 9 Maths Chapter 8 Predicting What Comes Next Exploring Sequences and Progressions?

They are the complete, step-by-step answers to all the exercise and in-text questions of Chapter 8 Predicting What Comes Next Exploring Sequences and Progressions from the NCERT Class 9 Maths textbook Ganita Manjari, written by experts as per the latest NCERT syllabus.

How can I download the Class 9 Maths Chapter 8 solutions PDF for free?

Open this page on aglasem, read the Predicting What Comes Next Exploring Sequences and Progressions questions with answers, and click the “Download Solutions PDF” link. The Class 9 Maths Chapter 8 NCERT Solutions PDF is completely free to download.

Are these NCERT Solutions as per the latest 2026-27 syllabus?

Yes. The NCERT Solutions for Class 9 Maths Chapter 8 are based on the latest NCERT textbook Ganita Manjari and the current 2026-27 CBSE syllabus, so the questions and answers match what you study in class.

Where can I get NCERT Solutions for the other chapters of Class 9 Maths?

You can find the answers to every chapter on the NCERT Solutions for Class 9 Maths page, and solutions for every subject on the NCERT Solutions for Class 9 page.

How do NCERT Solutions help in exam preparation?

They show the correct method to solve each question, help you write answers the way they are expected in exams, let you check and correct your own work, and save revision time — which together improve your marks in Class 9 Maths.

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