Sequences and Proportion: Practice Questions

These are original questions written for Marlbridge, in the style and at the standard of the examination. They are not reproduced past-paper questions — examination boards hold copyright in their own papers. Use these alongside the official past papers available free from your board.

Related: Sequences and Proportion revision notes, covering subtopics 2.7 Sequences and 2.8 Proportion for Cambridge O Level Mathematics (Syllabus D) 4024.

---

Section A

1. Find the nth term of the sequence 5, 9, 13, 17, … [2]

2. Find the next two terms of the sequence 3, 6, 12, 24, … and state its type. [2]

Section B

3. A sequence has nth term 3n² − 1.

(a) Find the first four terms. [2] (b) Determine whether 107 is a term of the sequence. [3]

4. Find the nth term of the quadratic sequence 4, 11, 22, 37, 56, … [4]

5. y is directly proportional to x. When x = 8, y = 20.

(a) Find the equation connecting y and x. [3] (b) Find y when x = 14. [1] (c) Find x when y = 42.5. [2]

6. p is inversely proportional to the square of q. When q = 3, p = 8.

(a) Find the equation connecting p and q. [3] (b) Find p when q = 6. [2] (c) Explain what happens to p when q is doubled. [2]

7. Find the nth term of the geometric sequence 6, 3, 1.5, 0.75, … [2]

8. y is directly proportional to x³. When x = 2, y = 40.

(a) Find the equation connecting y and x. [3] (b) Find y when x = 5. [1]

9. Find the nth term of the cubic sequence 1, 8, 27, 64, … and hence state its 7th term. [3]

10. y is directly proportional to the square root of x. When x = 16, y = 12.

(a) Find the equation connecting y and x. [3] (b) Find y when x = 25. [1]

---

Answers

1. The common difference is 4 [1]; nth term = 4n + 1 [1].

2. 48 and 96 [1]; it is a geometric sequence with common ratio 2 [1].

3. (a) 2, 11, 26, 47 [1] [1]. (b) 3n² − 1 = 107, so 3n² = 108 [1] and n² = 36, giving n = 6 [1]. Since n is a positive integer, 107 is the 6th term [1].

4. First differences: 7, 11, 15, 19 [1]. Second differences: 4, so the coefficient of n² is 4 ÷ 2 = 2 [1]. Subtracting 2n² (2, 8, 18, 32, 50) gives 2, 3, 4, 5, 6, whose nth term is n + 1 [1]. So the nth term is 2n² + n + 1 [1].

5. (a) y = kx [1]; 20 = 8k so k = 2.5 [1]; y = 2.5x [1]. (b) y = 2.5 × 14 = 35 [1]. (c) 42.5 = 2.5x [1]; x = 17 [1].

6. (a) p = k ÷ q² [1]; 8 = k ÷ 9 so k = 72 [1]; p = 72 ÷ q² [1]. (b) p = 72 ÷ 36 [1] = 2 [1]. (c) q² becomes four times larger [1], so p becomes one quarter of its original value [1].

7. First term a = 6, common ratio r = 0.5 [1]; nth term = a × rⁿ⁻¹ = 6 × 0.5ⁿ⁻¹ [1].

8. (a) y = kx³ [1]; 40 = k(2³) = 8k, so k = 5 [1]; y = 5x³ [1]. (b) y = 5 × 5³ = 5 × 125 = 625 [1].

9. These are the cube numbers, so the nth term is n³ [1]. Checking: 1³=1, 2³=8, 3³=27, 4³=64 ✓ [1]. The 7th term is 7³ = 343 [1].

10. (a) y = k√x [1]; 12 = k√16 = 4k, so k = 3 [1]; y = 3√x [1]. (b) y = 3 × √25 = 3 × 5 = 15 [1].

---

Where marks are usually lost

• Writing the nth term as “4n” without the constant.
• Forgetting to halve the second difference for the n² coefficient.
• Not finding k before substituting in proportion questions.
• Forgetting that inverse square proportion means quartering, not halving.
• Confusing y ∝ x³ with y ∝ 3x — cube proportion means the equation contains x³, not a coefficient of 3.
• Not recognising a cubic sequence (1, 8, 27, 64, …) as the cube numbers, and instead trying to force a quadratic nth-term method onto it.

Recognising a sequence type quickly

Sequence type	Signal	nth term form
Linear	Constant first difference	an + b
Quadratic	Constant second difference	an² + bn + c
Cubic	Terms match n³, 2n³, etc. (check against cube numbers)	an³ + …
Exponential (geometric)	Constant ratio between terms	a × rⁿ⁻¹

Always write out the differences (and, if those aren’t constant, the ratios) underneath the sequence before deciding which type you are dealing with — committing to the wrong method part-way through a question is one of the most time-costly mistakes in this topic. For the full method for each sequence type and further worked examples of non-linear proportion, see the Sequences and Proportion revision notes and the linked study guide.