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Grade 6 · Higher · free GCSE maths worksheet

Geometric and Fibonacci-type sequences

Common ratio and recurrence rules.

13 original questions, 41 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Here are the first four terms of a sequence.

3    6    12    24

  1. Write down the next term of the sequence.
  2. Write down the term-to-term rule of the sequence.
2 marks
Show answer
a) 48 b) multiply by 2 (accept double the previous term)
2

Here are the first five terms of a Fibonacci sequence.

2    3    5    8    13

  1. Write down the next term of the sequence.
  2. Describe the rule for finding the next term of the sequence.
2 marks
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a) 21 b) open: add the two previous terms together
3

Here are the first four terms of a geometric sequence.

3    12    48    192

  1. Write down the common ratio.
  2. Work out the next term.
2 marks
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a) 4 b) 768
4

Here are the first four terms of a sequence.

96    48    24    12

  1. Write down the next term of the sequence.
  2. Work out the 8th term of the sequence.
3 marks
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a) 6 b) 0.75 (accept 3/4)
5

Here are the first four terms of a geometric sequence.

243    81    27    9

  1. Write down the common ratio.
  2. Work out the 5th term and the 6th term.
3 marks
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a) 1/3 b) 5th term 3, 6th term 1
6

Here are the first four terms of a geometric sequence.

3    6    12    24

  1. Write an expression for the nth term of the sequence.
  2. Work out the 8th term.
3 marks
Show answer
a) 3 × 2(n-1) b) 384
7

Here are three sequences.

Sequence A:  7    11    15    19

Sequence B:  2    6    18    54

Sequence C:  1    4    5    9    14

  1. Write down the letter of the geometric sequence.
  2. Work out the next term of sequence C.
  3. Work out the 10th term of sequence A.
3 marks
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a) B b) 23 c) 43
8

The first two terms of a Fibonacci-type sequence are 4 and 7. Each term after the second is the sum of the two terms before it.

  1. Write down the next three terms.
  2. Another Fibonacci-type sequence starts x, 3, ... Show that its 4th term is x + 6.
4 marks
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a) 11, 18, 29 b) x, 3, x + 3, x + 6, so the 4th term is x + 6
9

Here are the first five terms of a Fibonacci-type sequence. Each term after the second is the sum of the two terms before it.

3    x    ___    17    ___

Find the value of x and the 5th term.

3 marks
Show answer
x = 7, 5th term = 27
10

In a Fibonacci sequence, each term after the second is the sum of the two previous terms.

The first term of a Fibonacci sequence is p.

The second term of the sequence is 5.

The fourth term of the sequence is 22.

Work out the value of p.

3 marks
Show answer
p = 12
11

The first two terms of a geometric sequence are

5    5√2

  1. Work out the 4th term of the sequence.
    Give your answer in the form a√2 where a is an integer.
  2. Find the first term of the sequence that is greater than 100.
5 marks
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a) 10√2 b) 80√2 (the 10th term, approximately 113.1)
12

The first two terms of a Fibonacci-type sequence are 2 and n. Each term after the second is the sum of the two terms before it.

  1. Write expressions, in terms of n, for the 3rd, 4th and 5th terms.
  2. The 5th term is 31. Work out the value of n.
4 marks
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a) 2 + n, 2 + 2n, 4 + 3n b) n = 9
13

The first term of a geometric sequence is 2 and the 3rd term is 50. The common ratio is positive.

  1. Work out the common ratio.
  2. Work out the 6th term.
4 marks
Show answer
a) 5 b) 6250

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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