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About Sequences & Series — Class 9 IB

Identify arithmetic and geometric sequences; find nth terms and partial sums. This topic is part of the IB Class 9 mathematics syllabus (chapter: Unit 14). On this page you can practice 52 questions across three difficulty levels — 20 easy, 20 medium, and 12 hard — each with a visual step-by-step solution, plus a timed 29-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Sequences & Series — solved examples for Class 9 IB

Example 1easy

Which of the following statements best describes a mathematical sequence?
  1. A)A list of random numbers without any specific order.
  2. B)An ordered list of numbers that follow a specific rule or pattern.
  3. C)A set of numbers where each number is larger than the previous one.
  4. D)A collection of numbers that can be added together to find a total.

Step-by-step solution

  1. A sequence is fundamentally an ordered list of numbers.
  2. The key characteristic of a mathematical sequence is that its terms usually follow a specific rule or pattern, allowing you to predict subsequent terms.
  3. Options A, C, and D do not fully capture the essence of a mathematical sequence, especially the 'rule or pattern' aspect.

Answer: An ordered list of numbers that follow a specific rule or pattern.

Example 2medium

What type of sequence is 2, 6, 18, 54, ...?
  1. A)Arithmetic sequence
  2. B)Neither arithmetic nor geometric
  3. C)Both arithmetic and geometric
  4. D)Geometric sequence

Step-by-step solution

  1. First, check for a common difference (d): 6 - 2 = 4, but 18 - 6 = 12. Since the difference is not constant, it's not an arithmetic sequence.
  2. Next, check for a common ratio (r): 6 / 2 = 3, 18 / 6 = 3, 54 / 18 = 3. Since there is a constant ratio, it is a geometric sequence.

Answer: Geometric sequence

Example 3hard

Find the sum of all natural numbers between 100 and 500 (inclusive) that are divisible by 7 but not by 3.
  1. A)11571
  2. B)12100
  3. C)13000
  4. D)14500

Step-by-step solution

  1. Step 1: Find the sum of numbers divisible by 7 between 100 and 500. The first term (u₁) is 105 (7 × 15) and the last term (u_n) is 497 (7 × 71). The number of terms (n) = 71 - 15 + 1 = 57. The sum S_7 = n/2 (u₁ + u_n) = 57/2 (105 + 497) = 57/2 (602) = 57 × 301 = 17157.
  2. Step 2: Find the sum of numbers divisible by 21 (both 7 and 3) between 100 and 500. The first term (u₁) is 105 (21 × 5) and the last term (u_n) is 483 (21 × 23). The number of terms (n) = 23 - 5 + 1 = 19. The sum S_21 = n/2 (u₁ + u_n) = 19/2 (105 + 483) = 19/2 (588) = 19 × 294 = 5586.
  3. Step 3: Subtract the sum of numbers divisible by 21 from the sum of numbers divisible by 7. Required Sum = S_7 - S_21 = 17157 - 5586 = 11571.

Answer: 11571

Practice questions on Sequences & Series

  1. Q1.easy

    Maya observed a pattern of dots: 3 dots, then 6 dots, then 9 dots, then 12 dots. If this pattern continues, what type of sequence does the number of dots form?
    1. A)Geometric sequence because the number of dots is increasing.
    2. B)Arithmetic sequence because there is a constant difference between consecutive terms.
    3. C)Neither arithmetic nor geometric sequence because the pattern is too simple.
    4. D)Both arithmetic and geometric sequence because it involves addition and multiplication.
    Show answer

    Answer: Arithmetic sequence because there is a constant difference between consecutive terms.

    Hint: Examine the relationship between each term and the next. Is it adding a constant value, or multiplying by a constant value?

  2. Q2.easy

    Consider the sequence: 2, 6, 18, 54, ... Which statement correctly identifies the type of sequence and its defining characteristic?
    1. A)It is an arithmetic sequence with a common difference of 4.
    2. B)It is a geometric sequence with a common ratio of 3.
    3. C)It is an arithmetic sequence with a common difference of 3.
    4. D)It is a geometric sequence with a common ratio of 4.
    Show answer

    Answer: It is a geometric sequence with a common ratio of 3.

    Hint: Check if there's a constant value being added or a constant factor being multiplied to get from one term to the next.

  3. Q3.easy

    Ravi was asked to find the common difference for the sequence 15, 12, 9, 6, ... He calculated the common difference as 3. Which statement correctly identifies his error, if any?
    1. A)Ravi made no error; the common difference is indeed 3.
    2. B)Ravi made an error; the common difference should be -3.
    3. C)Ravi made an error; this is a geometric sequence, so there is no common difference.
    4. D)Ravi made an error; the common difference should be 15 - 6 = 9.
    Show answer

    Answer: Ravi made an error; the common difference should be -3.

    Hint: Remember that the common difference is found by subtracting a term from its *succeeding* term. Pay close attention to the sign.

  4. Q4.medium

    Find the 15th term of the arithmetic sequence: 5, 8, 11, 14, ...
    1. A)44
    2. B)47
    3. C)50
    4. D)53
    Show answer

    Answer: 47

    Hint: Recall the formula for the nth term of an arithmetic sequence: a_n = a + (n-1)d.

  5. Q5.medium

    The height of a plant at the end of each week forms an arithmetic progression. If it is 10 cm tall at the end of Week 1, 13 cm at the end of Week 2, and 16 cm at the end of Week 3, what will be its height at the end of Week 8?
    1. A)28 cm
    2. B)31 cm
    3. C)34 cm
    4. D)37 cm
    Show answer

    Answer: 31 cm

    Hint: First, determine the first term and common difference of the arithmetic progression representing the plant's height.

  6. Q6.medium

    Find the 6th term of the geometric sequence: 2, 6, 18, ...
    1. A)162
    2. B)243
    3. C)486
    4. D)729
    Show answer

    Answer: 486

    Hint: Recall the formula for the nth term of a geometric sequence: a_n = ar^(n-1).

  7. Q7.hard

    A ladder has rungs 25 cm apart. The bottom rung is 10 cm long, and the top rung is 50 cm long. If the rungs decrease uniformly in length and there are 21 rungs, what is the total length of wood required for all the rungs?
    1. A)580 cm
    2. B)600 cm
    3. C)630 cm
    4. D)650 cm
    Show answer

    Answer: 630 cm

    Hint: The lengths of the rungs form an arithmetic progression. Identify the first term, the last term, and the number of terms, then use the appropriate sum formula.

  8. Q8.hard

    Consider three positive numbers `a, b, c`. If they form an Arithmetic Sequence (AS) and also a Geometric Sequence (GS), which of the following statements must be true?
    1. A)a = b = c
    2. B)b = a + c
    3. C)b² = a × c
    4. D)a, b, c cannot exist simultaneously
    Show answer

    Answer: a = b = c

    Hint: Use the definitions for arithmetic and geometric sequences simultaneously: `2b = a + c` and `b² = ac`. Substitute one equation into the other and simplify.

  9. Q9.hard

    A sequence is defined by `a_1 = 1`, `a_2 = 2`, and `a_n = a_{n-1} + a_{n-2} + n` for `n ≥ 3`. Find the value of `a_5`.
    1. A)20
    2. B)23
    3. C)26
    4. D)29
    Show answer

    Answer: 23

    Hint: Work step-by-step from the given initial terms, applying the recursive formula for each subsequent term until you reach `a_5`.

These are 9 of the 52 questions available for Sequences & Series. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.