AQA GCSE Maths · 3.2.4

Sequences

50 maths questions with worked answers with answers for 3.2.4 Sequences in AQA GCSE Maths (spec 8300). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official AQA material.

All 50 questions and answers

Question 1 · Arithmetic · A23
Give the next term: 4, 7, 10, 13, ...
Answer:
The common difference is 3.
13 + 3 = 16.
Question 2 · Arithmetic · A23
Give the next two terms: 20, 16, 12, 8, ...
Answer:
Subtract 4 each time.
8 − 4 = 4, then 4 − 4 = 0.
Question 3 · Arithmetic · A23
The first term is 6 and each term increases by 5. Find the fourth term.
Answer:
From the first to the fourth term there are 3 increases.
6 + 3 × 5 = 21.
Question 4 · Arithmetic · A23
Find the 10th term of 3, 7, 11, 15, ...
Answer:
The common difference is 4; the 10th term is 9 steps after the first.
3 + 9 × 4 = 39.
Question 5 · Arithmetic · A25
Find the nth term of 5, 8, 11, 14, ...
Answer:
The common difference is 3, so begin with 3n.
At n = 1, 3n = 3; add 2 to obtain 5.
The nth term is 3n + 2.
Question 6 · Arithmetic · A25
Find the nth term of 12, 9, 6, 3, ...
Answer:
The common difference is −3, so begin with −3n.
At n = 1, add 15 to −3 to obtain 12.
The nth term is 15 − 3n.
Question 7 · Arithmetic · A25
Find the nth term of −1, 3, 7, 11, ...
Answer:
The common difference is 4, so begin with 4n.
At n = 1, subtract 5 from 4 to obtain −1.
The nth term is 4n − 5.
Question 8 · Arithmetic · A25
The nth term is 6n − 1. Find the fifth term.
Answer:
Substitute n = 5.
6 × 5 − 1 = 29.
Question 9 · Arithmetic · A25
Is 62 a term of the sequence 4n + 2 for positive integers n?
Answer:
Set 4n + 2 = 62.
4n = 60, so n = 15, a positive integer.
Yes, it is a term.
Question 10 · Arithmetic · A25
Is 50 a term of the sequence 3n + 1 for positive integers n?
Answer:
Set 3n + 1 = 50.
3n = 49, so n = 493, not an integer.
No, it is not a term.
Question 11 · Arithmetic · A25
Find the first term of the sequence 7n − 10.
Answer:
Substitute n = 1.
7 × 1 − 10 = −3.
Question 12 · Arithmetic · A25
What is the common difference for the sequence 2n + 9?
Answer:
Increasing n by 1 increases 2n + 9 by 2.
The common difference is 2.
Question 13 · Arithmetic · A25
Find n when the term 5n + 3 equals 43.
Answer:
Set 5n + 3 = 43.
5n = 40, so n = 8.
Question 14 · Arithmetic · A25
The nth term is 10 − 2n. Which term first equals 0?
Answer:
Set 10 − 2n = 0.
2n = 10, so n = 5: the fifth term.
Question 15 · Arithmetic · A23
A sequence starts at 5. Each term is 3 more than the previous one. Give term 6.
Answer:
The sixth term is 5 steps after the first.
5 + 5 × 3 = 20.
Question 16 · Squares · A24
Give the next square number after 81.
Answer:
81 = 9².
The next square is 10² = 100.
Question 17 · Squares · A24
What is the seventh square number?
Answer:
The seventh square number is 7².
7 × 7 = 49.
Question 18 · Squares · A24
Find the difference between 9² and 8².
Answer:
9² = 81 and 8² = 64.
81 − 64 = 17.
Question 19 · Cubes · A24
Give the next cube number after 27.
Answer:
27 = 3³.
The next cube is 4³ = 64.
Question 20 · Cubes · A24
What is the fifth cube number?
Answer:
The fifth cube number is 5³.
5 × 5 × 5 = 125.
Question 21 · Triangular numbers · A24
Give the next term: 1, 3, 6, 10, ...
Answer:
The differences are 2, 3 and 4; the next difference is 5.
10 + 5 = 15.
Question 22 · Triangular numbers · A24
What is the sixth triangular number?
Answer:
Add the first six positive integers.
1 + 2 + 3 + 4 + 5 + 6 = 21.
Question 23 · Triangular numbers · A24
Why is the fourth triangular number 10?
Answer:
The first four rows have 1, 2, 3 and 4 dots.
1 + 2 + 3 + 4 = 10.
Question 24 · Fibonacci-type · A24
A sequence starts 2, 5; each later term is the sum of the previous two. Give the next three terms.
Answer:
Add consecutive pairs: 2 + 5 = 7.
Then 5 + 7 = 12 and 7 + 12 = 19.
Next terms: 7, 12, 19.
Question 25 · Fibonacci-type · A24
A sequence starts 7, −2; each later term is the sum of the previous two. Give the next two terms.
Answer:
7 + (−2) = 5.
Then −2 + 5 = 3.
Next terms: 5, 3.
Question 26 · Fibonacci-type · A24
The terms are 3, 4, 7, 11, ... Find the next term.
Answer:
Add the previous two terms.
7 + 11 = 18.
Question 27 · Geometric · A24
Give the next term: 3, 6, 12, 24, ...
Answer:
Each term is twice the previous term.
24 × 2 = 48.
Question 28 · Geometric · A24
Find the common ratio of 5, 15, 45, 135, ...
Answer:
Divide a term by the previous term.
15 ÷ 5 = 3, so the common ratio is 3.
Question 29 · Geometric · A24
Give the next term: 64, 32, 16, 8, ...
Answer:
Each term is half the previous term.
8 ÷ 2 = 4.
Question 30 · Geometric · A24
Find the fifth term of 2, 6, 18, 54, ...
Answer:
Each term is multiplied by 3.
54 × 3 = 162.
Question 31 · Geometric · A24
A sequence has first term 4 and multiplies by 12 each time. Find term 4.
Answer:
Multiply by 12 at each step: 4, 2, 1, 12.
The fourth term is 12.
Question 32 · Geometric · A24
Find the nth term of 3, 6, 12, 24, ...
Answer:
The first term is 3, and each step multiplies by 2.
The nth term is 3 × 2^{n − 1}, since there are n − 1 steps.
Question 33 · Geometric · A24 · Higher only
A sequence starts 1 and has common ratio √2. Give the third term.
Answer:
First term = 1; second = √2.
Third = √2 × √2 = 2.
Question 34 · Quadratic · A24
Give the next term: 2, 5, 10, 17, ...
Answer:
The differences are 3, 5 and 7; the next is 9.
17 + 9 = 26.
Question 35 · Quadratic · A24
Find the second difference of 3, 8, 15, 24, ...
Answer:
First differences are 5, 7 and 9.
Their differences are 2 and 2, so the second difference is 2.
Question 36 · Quadratic · A24
The nth term is n² + 1. Find the fourth term.
Answer:
Substitute n = 4: 4² + 1.
16 + 1 = 17.
Question 37 · Quadratic · A25 · Higher only
Find the nth term of 2, 5, 10, 17, ...
Answer:
Compare the terms with 1², 2², 3², 4²: 1, 4, 9, 16.
Each given term is 1 more, so the nth term is n² + 1.
Question 38 · Quadratic · A25 · Higher only
Find the nth term of 1, 4, 9, 16, ...
Answer:
The terms are 1², 2², 3², 4².
The nth term is n².
Question 39 · Quadratic · A25 · Higher only
Find the nth term of 3, 6, 11, 18, ...
Answer:
Compare with square numbers 1, 4, 9, 16.
Each term is 2 more, so the nth term is n² + 2.
Question 40 · Quadratic · A25 · Higher only
Find the nth term of 2, 6, 12, 20, ...
Answer:
Compare the terms with n²: 1, 4, 9, 16.
The extra amounts are 1, 2, 3, 4, which equal n.
The nth term is n² + n.
Question 41 · Quadratic · A24
Find the next term: 4, 9, 16, 25, ...
Answer:
The terms are 2², 3², 4², 5².
The next is 6² = 36.
Question 42 · Patterns · A23
The nth term is 2n + 3. Find the first three terms.
Answer:
Substitute n = 1, 2 and 3 into 2n + 3.
The terms are 5, 7 and 9.
Question 43 · Patterns · A23
A pattern has 4 tiles in stage 1 and 3 extra tiles per new stage. How many tiles are in stage 5?
Answer:
Stage 5 is 4 stages after stage 1.
4 + 4 × 3 = 16 tiles.
Question 44 · Patterns · A25
A pattern has 7 tiles at stage 1 and 5 extra tiles per stage. Write the nth term.
Answer:
The increase is 5 each stage, so begin with 5n.
At n = 1, add 2 to obtain 7.
The nth term is 5n + 2.
Question 45 · Patterns · A23
A sequence starts at 10; add 2, then subtract 1, and repeat. Give the next four terms.
Answer:
Apply the repeating changes: 10 + 2 = 12; 12 − 1 = 11.
Then 11 + 2 = 13; 13 − 1 = 12.
Next terms: 12, 11, 13, 12.
Question 46 · Patterns · A25
What is the 20th term of the sequence 5n − 2?
Answer:
Substitute n = 20.
5 × 20 − 2 = 98.
Question 47 · Patterns · A23
A sequence starts 1, 4, 7, 10. How many must be added each time?
Answer:
Compare consecutive terms: 4 − 1 = 3 and 7 − 4 = 3.
Add 3 each time.
Question 48 · Patterns · A23
The nth term is n² − n. Find term 5.
Answer:
Substitute n = 5: 5² − 5.
25 − 5 = 20.
Question 49 · Error check · A25
A student says the nth term of 4, 7, 10, ... is 3n + 4. Correct it.
Answer:
The difference is 3, so begin with 3n.
At n = 1, add 1 to obtain 4.
The correct nth term is 3n + 1.
Question 50 · Error check · A23
A student calls 2, 4, 8, 16 arithmetic. What type is it?
Answer:
Each term is twice the previous term, so the common ratio is 2.
The differences 2, 4 and 8 are not constant.
It is geometric.

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