A23–A25Sequences and the nth term

Edexcel GCSE Maths (1MA1), Higher tier · Algebra

Practise Sequences and the nth term. 1 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.

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Sample questions

Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.

Question 1Easy3 marks

Here are the first five terms of an arithmetic sequence.
\(54,\quad 50,\quad 46,\quad 42,\quad 38\)

(a) Write down the next term of the sequence.[1]

(b) Find an expression, in terms of \(n\), for the \(n\)th term of the sequence.[2]

Show the answer and mark scheme

(a) Answer: \(34\)

B1 for 34

Worked solution: \(38 - 4 = 34\)

(b) Answer: \(58 - 4n\)

M1 for \(-4n + k\) for any value of \(k\), or a common difference of \(-4\) used
A1 for \(58 - 4n\) oe

Worked solution: The common difference is \(-4\), so the \(n\)th term is \(-4n + k\).
When \(n = 1\): \(-4 + k = 54\) so \(k = 58\). \(n\)th term \(= 58 - 4n\).

Question 2Medium4 marks

Here are the first five terms of an arithmetic sequence.
\(29,\quad 23,\quad 17,\quad 11,\quad 5\)

(a) Find an expression, in terms of \(n\), for the \(n\)th term of the sequence.[2]

(b) Is \(-358\) a term of this sequence?
You must show how you get your answer.[2]

Show the answer and mark scheme

(a) Answer: \(35 - 6n\)

M1 for \(-6n + k\) for any value of \(k\), or a common difference of \(-6\) used
A1 for \(35 - 6n\) oe

Worked solution: The common difference is \(-6\), so the \(n\)th term is \(-6n + k\).
When \(n = 1\): \(-6 + k = 29\) so \(k = 35\). \(n\)th term \(= 35 - 6n\).

(b) Answer: No: \(n = 65.5\) is not a whole number, so −358 is not a term.

M1 for \(35 - 6n = -358\) or a correct method to find the terms either side of −358
C1 for no, with a correct reason, e.g. \(n = 65.5\) is not an integer, or the 65th and 66th terms are −355 and −361

Worked solution: \(35 - 6n = -358\) gives \(-6n = -393\) so \(n = 65.5\).
No: \(n = 65.5\) is not a whole number, so −358 is not a term.

Question 3Hard4 marks

Sequence P has \(n\)th term \(5n + 2\).
Sequence Q has \(n\)th term \(3n + 10\).

(a) Find the three smallest numbers that are in both sequences.[2]

(b) Find an expression for the \(n\)th term of the sequence of numbers that are in both P and Q.
Give a reason for the common difference of this sequence.[2]

Show the answer and mark scheme

(a) Answer: 22, 37, 52

M1 for listing terms of both sequences: 7, 12, 17, 22, 27, ... and 13, 16, 19, 22, 25, ...
A1 for 22, 37 and 52

Worked solution: P: 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, ...
Q: 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, ...
Common: 22, 37, 52.

(b) Answer: \(15n + 7\): the common difference is 15 because 15 is the lowest common multiple of 5 and 3.

B1 for \(15n + 7\) oe
C1 for the reason: the terms of P go up in 5s and the terms of Q in 3s, so the common terms repeat every LCM(5, 3) = 15

Worked solution: After 22, the next common number is 15 more, because 15 is the smallest number that is a multiple of both 5 and 3.
The common numbers are 22, 37, 52, ... with first term 22 and difference 15: \(15n + 7\).

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