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Grade 7 · Higher · free GCSE maths worksheet

Arithmetic series: nth term and sum

Edexcel IGCSE

a + (n - 1)d and the sum of the first n terms.

10 original questions, 35 marks. Type your answers below for an instant mark and a score, or 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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How to do it

  1. Write down a (the first term) and d (the gap from one term to the next)
  2. nth term = a + (n - 1)d: count the gaps, not the terms
  3. Sum of n terms: Sn = n/2 [2a + (n - 1)d], or n/2 (a + l) if you know the last term l
  4. Two terms given? Write a + (n - 1)d for each and solve the simultaneous equations
  5. Given the sum, set Sn equal to it and solve the quadratic: keep the positive n

Watch out: The 20th term has 19 gaps after the first term, so it is a + 19d, not a + 20d.

Worked example

The first three terms of an arithmetic series are

5 + 9 + 13 + ...

Find the sum of the first 30 terms of the series.

a = 5, d = 4, n = 30

Formula sheet: Sn = n2[2a + (n - 1)d]

S30=302[2 × 5 + 29 × 4]
=15 × [10 + 116]
=15 × 126

1890

Check with Sn = n2(a + l), last term l = 5 + 29 × 4 = 121

S30=15 × (5 + 121) = 1890

The questions

1

Here is an arithmetic series.

6 + 11 + 16 + 21 + 26 + ...

  1. Write down the first term, a.
  2. Write down the common difference, d.
2 marks
Show answer
a) a = 6 b) d = 5
2

An arithmetic sequence has first term 3 and common difference 7.

Find the 20th term of the sequence.

2 marks
Show answer
136
3

The first four terms of an arithmetic sequence are

40,   37,   34,   31,   ...

Find the 25th term of the sequence.

2 marks
Show answer
-32
4

The first four terms of an arithmetic series are

3 + 8 + 13 + 18 + ...

Find the sum of the first 40 terms of the series.

You may use Sn = n2[2a + (n − 1)d]

3 marks
Show answer
4020
5

In an arithmetic sequence, the 3rd term is 11 and the 8th term is 31.

  1. Find the first term, a.
  2. Find the common difference, d.
  3. Find the sum of the first 20 terms of the sequence.
5 marks
Show answer
a) a = 3 b) d = 4 c) 820
6

A theatre has 25 rows of seats.

There are 18 seats in the first row, 21 seats in the second row, 24 seats in the third row, and so on. Each row has 3 more seats than the row in front of it.

  1. Work out the number of seats in the back row.
  2. Work out the total number of seats in the theatre.
4 marks
Show answer
a) 90 b) 1350
7

The first three terms of an arithmetic sequence are

7,   12,   17,   ...

  1. Find the value of n for which the nth term is the first term of the sequence that is greater than 300.
  2. Write down the value of this term.
3 marks
Show answer
a) n = 60 b) 302
8

Priya is saving for a bicycle.

She saves £5 in the first week, £7 in the second week, £9 in the third week, and so on. Each week she saves £2 more than the week before.

  1. Work out the amount, in pounds, that Priya saves in week 12.
  2. Work out the total amount, in pounds, that Priya has saved by the end of week 12.
  3. Priya needs more than £1000 for the bicycle. Work out the number of weeks it takes for her total savings to first exceed £1000.
5 marks
Show answer
a) 27 b) 192 c) 30
9

The first three terms of an arithmetic series are

5 + 8 + 11 + ...

The sum of the first n terms of the series is 1455.

Find the value of n.

4 marks
Show answer
n = 30
10

The first three terms of an arithmetic series are

4 + 10 + 16 + ...

  1. Find the 11th term of the series.
  2. Find the sum of the 11th term to the 30th term inclusive.
5 marks
Show answer
a) 64 b) 2420

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Every question here is original and written by Robinson Tuition. This topic is on the Pearson Edexcel International GCSE (4MA1) and is not examined on the UK GCSE; the grade is our estimate against the Edexcel grade descriptors. This is not an exam-board paper.

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