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Grade 4 · Foundation and Higher · free GCSE maths worksheet

Finding the nth term of a linear sequence

The rule an + b.

14 original questions, 43 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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The questions

1

Here are the first four terms of an arithmetic sequence.

5   9   13   17

Find an expression, in terms of n, for the nth term of the sequence.

2 marks
Show answer
4n + 1
2

Here are the first four terms of an arithmetic sequence.

20   17   14   11

Find an expression, in terms of n, for the nth term of the sequence.

2 marks
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23 - 3n
3

Here are the first four terms of an arithmetic sequence.

7   12   17   22

Explain why 81 is not a term of this sequence.

2 marks
Show answer
Every term of the sequence ends in 2 or 7, and 81 ends in 1
4

The nth term of a sequence is 6n − 1

Write down the first three terms of the sequence.

2 marks
Show answer
5, 11, 17
5

Here are the first four terms of an arithmetic sequence.

3   10   17   24

Find an expression, in terms of n, for the nth term of the sequence.

2 marks
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7n - 4
6

Matchsticks are used to make a row of squares. One square uses 4 matchsticks, two squares joined in a row use 7 matchsticks, and three squares joined in a row use 10 matchsticks. The pattern continues.

  1. Find an expression, in terms of n, for the number of matchsticks needed for n squares.
  2. How many matchsticks are needed for 20 squares?
3 marks
Show answer
a) 3n + 1 b) 61
7

Here are the first four terms of an arithmetic sequence.

4   7   10   13

  1. Write down the next term of the sequence.
  2. Find an expression, in terms of n, for the nth term of the sequence.
3 marks
Show answer
a) 16 b) 3n + 1
8

Here are the first four terms of a sequence.

7   11   15   19

  1. Find an expression, in terms of n, for the nth term.
  2. Which term in the sequence has a value of 99?
3 marks
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a) 4n + 3 b) 24
9

The nth term of a sequence is 5n − 3

  1. Work out the 12th term of the sequence.
  2. Is 122 a term of the sequence? You must show how you get your answer.
3 marks
Show answer
a) 57 b) Yes, 5n - 3 = 122 gives n = 25, a whole number
10

Here are the first four terms of an arithmetic sequence.

8   15   22   29

  1. Find an expression, in terms of n, for the nth term of the sequence.
  2. Explain why 150 is not a term of the sequence.
4 marks
Show answer
a) 7n + 1 b) 7n + 1 = 150 gives 7n = 149, and 149 is not divisible by 7
11

Here are the first three patterns in a sequence made from grey square tiles.

Pattern 1 uses 5 tiles. Pattern 2 uses 8 tiles. Pattern 3 uses 11 tiles.

The number of tiles goes up by the same amount from each pattern to the next.

  1. Work out the number of tiles in pattern 5.
  2. Write down an expression, in terms of n, for the number of tiles in pattern n.
  3. Explain why there is no pattern in this sequence that uses exactly 100 tiles.
Pattern 1Pattern 2Pattern 3
5 marks
Show answer
a) 17 b) 3n + 2 (accept 2 + 3n) c) open: 3n + 2 = 100 gives 3n = 98, which has no whole number solution (98 is not a multiple of 3)
12

The first term of an arithmetic sequence is 3.

The term-to-term rule is add 6.

  1. Work out the 5th term of the sequence.
  2. Find an expression, in terms of n, for the nth term of the sequence.
  3. Explain why every term of the sequence is an odd number.
4 marks
Show answer
a) 27 b) 6n - 3 c) 6n is even, so 6n - 3 is always odd
13

The nth term of a sequence is 50 − 6n.

  1. Write down the first three terms of the sequence.
  2. Work out the first term of the sequence that is negative.
4 marks
Show answer
a) 44, 38, 32 b) -4
14

Here are the first four terms of an arithmetic sequence.

100   93   86   79

  1. Write down the next two terms of the sequence.
  2. Find an expression, in terms of n, for the nth term of the sequence.
  3. Explain why 0 is not a term of the sequence.
4 marks
Show answer
a) 72 and 65 b) 107 - 7n c) 107 - 7n = 0 gives 7n = 107, and 107 is not divisible by 7

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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