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Class 9 Maths Chapter 8: Predicting What Comes Next: Exploring Sequences and Progressions — Important Questions & Sample Paper

CBSE· Ganita Manjari· 44+ original questions readyहिन्दी में देखें

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Reviewed by qpaper's CBSE curriculum team · Edited by Mohit · Updated June 2026

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Yes — this page has 44+ original Class 9 Mathematics Chapter 8 (“Predicting What Comes Next: Exploring Sequences and Progressions”) important questions with answers (Multiple Choice (MCQ), Assertion–Reason, Short Answer, Short Answer, Long Answer, Case Study). Practise them free, or generate a full CBSE board-pattern sample paper (80 marks) and export it to PDF or Word — in English & Hindi, for 2026-27.

Chapter 8 of NCERT's Ganita Manjari for Class 9, titled 'Predicting What Comes Next: Exploring Sequences and Progressions', introduces students to the fascinating world of number patterns and sequences. This chapter goes beyond simple counting; it develops the skill of recognizing underlying rules in ordered lists of numbers or shapes. Key concepts include arithmetic sequences (where a constant difference separates consecutive terms), recursive definitions (like the Virahānka–Fibonacci sequence where each term is the sum of the two before it), and explicit formulas that directly compute the nth term. Students learn to move between verbal descriptions, recursive rules, and algebraic expressions like tn = n² – n + 1. Visual patterns also play a role, such as triangular numbers T_n = n(n+1)/2 and problems involving hollow triangular arrangements, linking geometry to algebraic thinking. Exam questions typically ask students to identify patterns, write next terms, derive nth term formulas, or prove simple properties. You’ll encounter tasks like extending a sequence (2,5,8,11,…), calculating Fibonacci numbers, proving a recursive formula matches an explicit one, or exploring dot patterns. By mastering this chapter, students build a strong foundation for more advanced topics like arithmetic progressions and mathematical induction in higher classes.

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Mathematics — Predicting What Comes Next: Exploring Sequences and Progressions

Class 9Time: 3 hrsMax Marks: 80

SECTION A

  1. 1.

    In an arithmetic progression, the constant difference between consecutive terms is called the:

    (a) common ratio(b) common difference(c) common term(d) common sum
    1
  2. 2.

    Which sequence corresponds to the explicit formula p_n = n^2 + 2?

    (a) 3, 6, 11, 18, ...(b) 1, 4, 9, 16, ...(c) 2, 5, 10, 17, ...(d) 3, 5, 7, 9, ...
    1
  3. 3.

    The 12th term of an arithmetic progression with first term -8 and common difference 5 is:

    (a) 47(b) 52(c) 57(d) 42
    1

+ 41 more questions in the full paper

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Marks distribution & blueprint

In a CBSE exam, this chapter typically contributes questions across the following types. The last column shows how many original questions of each type we have ready in our bank for this chapter:

Question typeMarks eachIn our bank
Multiple Choice (MCQ)1 mark13
Assertion–Reason1 mark6
Short Answer2 marks8
Short Answer3 marks6
Long Answer5 marks5
Case Study4 marks6

44 original, exam-style questions in our bank for this chapter — with answers.

Important & sample questions (with answers)

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  1. Q1. In an arithmetic progression, the constant difference between consecutive terms is called the:

    1 mark
    Multiple Choice (MCQ)
    (A) common ratio(B) common difference(C) common term(D) common sum
    ▸ Answer

    common difference

  2. Q2. Which sequence corresponds to the explicit formula p_n = n^2 + 2?

    1 mark
    Multiple Choice (MCQ)
    (A) 3, 6, 11, 18, ...(B) 1, 4, 9, 16, ...(C) 2, 5, 10, 17, ...(D) 3, 5, 7, 9, ...
    ▸ Answer

    3, 6, 11, 18, ...

  3. Q3. The 12th term of an arithmetic progression with first term -8 and common difference 5 is:

    1 mark
    Multiple Choice (MCQ)
    (A) 47(B) 52(C) 57(D) 42
    ▸ Answer

    47

  4. Q4. A sequence is defined recursively as b_1 = 3, b_n = b_{n-1} + 4 for n ≥ 2. What is b_4?

    1 mark
    Multiple Choice (MCQ)
    (A) 15(B) 19(C) 11(D) 23
    ▸ Answer

    15

  5. Q5. Assertion (A): The triangular numbers 1, 3, 6, 10, 15, ... can be generated by the recursive rule t1 = 1, tn = tn-1 + n for n ≥ 2. Reason (R): The nth triangular number is the sum of the first n natural numbers.

    1 mark
    Assertion–Reason
    (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.
    ▸ Answer

    Both A and R are true and R is the correct explanation of A.

  6. Q6. The nth term of a sequence is given by the formula tn = 3n + 2. Find the 10th term.

    2 marks
    Short Answer
    ▸ Answer

    32

  7. Q7. The Virahānka–Fibonacci sequence is defined by F1 = 1, F2 = 1, and Fn = Fn-1 + Fn-2 for n ≥ 3. Calculate the value of F6.

    2 marks
    Short Answer
    ▸ Answer

    8

  8. Q8. A sequence is defined recursively as T1 = 1, Tn = 2T(n-1) + 1 for n ≥ 2. Prove by induction that the explicit formula for the nth term is Tn = 2^n - 1.

    3 marks
    Short Answer
    ▸ Answer

    Proof by induction: Base case n=1 holds. Assuming Tk = 2^k - 1, then T(k+1) = 2(2^k - 1) + 1 = 2^(k+1) - 1.

  9. Q9. A sequence is generated using the recursive rule: a1 = 4, an = a(n-1) + 7 for n ≥ 2. Write the first six terms of this sequence.

    3 marks
    Short Answer
    ▸ Answer

    4, 11, 18, 25, 32, 39

  10. Q10. The nth term of a sequence is given by s_n = n^2 + 2n + 1. (i) Write down the first four terms of the sequence. (ii) Express s_n as a perfect square. (iii) Consider the sequence of differences d_n = s_{n+1} - s_n. Find the first three terms of this new sequence and show that it forms an arithmetic progression. State its first term and common difference.

    5 marks
    Long Answer
    ▸ Answer

    (i) 4, 9, 16, 25. (ii) s_n = (n+1)^2. (iii) d_n = 2n+3; first three terms: 5,7,9; it is an AP with first term 5 and common difference 2.

  11. Q11. A pattern of dots forms an equilateral triangular arrangement with n dots on each side. The total number of dots is the nth triangular number: T_n = n(n+1)/2. A 'hollow triangle' of side n (n ≥ 3) has dots only on the boundary; the interior is empty. (i) Find an expression for the number of boundary dots, B_n, in terms of n. (ii) Show that B_n can be written as a linear function of n. (iii) A hollow triangle of side n is placed inside a larger hollow triangle of side n+3. How many additional dots are on the boundary of the larger triangle compared to the smaller one? Express your answer in terms of n.

    5 marks
    Long Answer
    ▸ Answer

    (i) B_n = 3n - 3. (ii) B_n = 3n - 3 is linear. (iii) 9.

  12. Q12. A ladder is designed with horizontal rungs placed at equal vertical spacings. The first rung is at a height of 30 cm from the bottom, the second rung at 60 cm, the third at 90 cm, and so on. The topmost rung is exactly at 300 cm from the bottom.

    4 marks
    Case Study
    1. (i) How many rungs does the ladder have?2 marks
    2. (ii) If the ladder is extended so that the top rung is at 390 cm, keeping the same spacing, how many additional rungs must be added?2 marks
    ▸ Answer

    The original ladder has 10 rungs. To reach 390 cm, 3 more rungs need to be added, making it 13 rungs.

Frequently asked questions

What are the important topics in Class 9 Mathematics Chapter 8 'Predicting What Comes Next'?

This chapter focuses on recognizing patterns in number sequences, including arithmetic sequences with a common difference, recursive sequences like the Virahānka–Fibonacci sequence, and visual patterns like triangular numbers. Key skills are identifying the rule, writing the next terms, and expressing the nth term either recursively or explicitly. Understanding both recursive definitions and explicit formulas is crucial.

What types of exam questions come from this chapter?

Typical exam questions ask students to extend a given sequence, find missing terms, identify the rule (e.g., 'add 3' or 'multiply by 2 and add 1'), switch between recursive and explicit forms, and solve problems involving triangular numbers or hollow triangular dot patterns. Proof-based questions, such as showing that the difference between consecutive terms is always even, also appear.

How can I get better at solving sequence and progression problems?

Practice is key. Start by carefully observing the given terms to find the pattern—look for constant differences, ratios, or relationships to previous terms. Write the rule in words first, then translate to symbols. Work through varied problems: arithmetic sequences, Fibonacci-type recursions, and shape-based patterns. Using a question bank like qpaper.in lets you generate customized practice sets to reinforce these concepts.

Is the Fibonacci sequence part of the CBSE Class 9 syllabus?

Yes, the chapter includes the Virahānka–Fibonacci sequence as an example of a recursively defined sequence. Students should know how to generate terms using Fn = Fn-1 + Fn-2 with given initial values, and may be asked to compute specific terms like F6. However, deep properties like the golden ratio are not required at this level.

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Class 9 Maths Ch8 — Important Questions & Sample Paper with Answers | Free PDF