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Grade 7 · Higher · free GCSE maths worksheet

nth term of a quadratic sequence

Using second differences.

15 original questions, 53 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 five terms of a quadratic sequence.

2   7   14   23   34

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

2 marks
Show answer
n2 + 2n - 1
2

The nth term of a sequence is n2 + 4.

  1. Write down the first three terms of the sequence.

  2. Is 85 a term of the sequence? You must show your working.

  3. Is 120 a term of the sequence? You must show your working.

4 marks
Show answer
a) 5, 8, 13 b) yes, the 9th term c) no
3

Here are the first five terms of a quadratic sequence.

5   11   19   29   41

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

2 marks
Show answer
n2 + 3n + 1
4

Here are the first five terms of a quadratic sequence.

1   10   25   46   73

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

3 marks
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3n2 - 2
5

A pattern of tiles is built in stages. Stage 1 uses 2 tiles, stage 2 uses 9 tiles, stage 3 uses 20 tiles and stage 4 uses 35 tiles. The number of tiles in each stage forms a quadratic sequence.

  1. Find an expression, in terms of n, for the number of tiles in stage n.

  2. Work out the number of tiles in stage 12.

  3. Mia says that one stage uses exactly 500 tiles. Is she correct? You must show your working.

5 marks
Show answer
a) 2n2 + n - 1 b) 299 c) no
6

Here are the first five terms of a quadratic sequence.

6   15   28   45   66

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

3 marks
Show answer
2n2 + 3n + 1
7

Here are the first five terms of a quadratic sequence.

0   7   18   33   52

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

3 marks
Show answer
2n2 + n - 3
8

Here are the first four terms of a quadratic sequence.

30    27    20    9

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

  2. Work out the value of the first term in the sequence that is negative.

4 marks
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a) -2n2 + 3n + 29 b) -6 (the 5th term)
9

Here are the first five terms of a quadratic sequence.

4   13   26   43   64

  1. Find an expression, in terms of n, for the nth term of this sequence.
  2. Hence find the 10th term of the sequence.

3 marks
Show answer
a) 2n2 + 3n - 1 b) 229
10

The first term of a quadratic sequence is 2. The differences between consecutive terms are 5, 9, 13, 17, ...
Find an expression, in terms of n, for the nth term of the sequence.

3 marks
Show answer
n-th term = 2n2 - n + 1
11

Here are the first five terms of a quadratic sequence.

20   38   52   62   68

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

3 marks
Show answer
-2n2 + 24n - 2
12

Here are the first five terms of a quadratic sequence.

3   8   14   21   29

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

3 marks
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n2 + 7n - 22
13

The nth term of a sequence is 2n2 + 5n.
Show that the difference between the (n + 1)th term and the nth term is 4n + 7.

3 marks
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4n + 7
14

The nth term of a quadratic sequence is an2 + bn, where a and b are constants.

The 2nd term of the sequence is 10.

The 4th term of the sequence is 36.

  1. Find an expression, in terms of n, for the nth term of the sequence.
  2. Find the value of n for which the nth term of the sequence is 105.

6 marks
Show answer
a) 2n2 + n b) 7
15

Here are the first five terms of a quadratic sequence.

7   16   29   46   67

  1. Find an expression, in terms of n, for the nth term of this sequence.
  2. Show that 500 is not a term of this sequence.

6 marks
Show answer
a) 2n2 + 3n + 2 b) 2n2 + 3n + 2 = 500 has no positive integer solution

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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