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Sequences

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Pearson Edexcel International GCSE Mathematics A (4MA1)

Sequences

The big idea: A sequence can be built step by step (term-to-term) or straight from the position number (position-to-term). Arithmetic sequences go up by the same amount every time, which is why their nth term is a linear expression.

Summary — What This Chapter Covers

Term-to-term rule: how to get from one term to the next. Position-to-term rule: the nth term formula.
Know the special sequences: odd, even, square, multiples and powers.
Arithmetic sequences have a common difference; the nth term is (difference)n + (adjustment).
Higher: first term a and common difference d; nth term a + (n − 1)d.
Higher: the sum of the first n terms, S_n = (n/2)[2a + (n − 1)d] — given on the formulae sheet.

1. Generating Sequences

Foundation & Higher

Term-to-term and position-to-term

A term-to-term rule needs a starting value: "first term 3, multiply by 2 then subtract 1". A position-to-term rule gives any term directly: substitute n = 1, 2, 3, … into the nth term. Recognise these:

Sequence	Terms	nth term
Even numbers	2, 4, 6, 8, …	2n
Odd numbers	1, 3, 5, 7, …	2n − 1
Square numbers	1, 4, 9, 16, …	n²
Multiples of 5	5, 10, 15, 20, …	5n
Powers of 2	2, 4, 8, 16, …	2ⁿ

Worked Example

Write down the first four terms of the sequence with first term 3 and term-to-term rule "multiply by 2 then subtract 1".

Practice Question 1

The nth term of a sequence is n² + 1. Write down the first three terms and the 10th term.

2. Continuing a Sequence

Foundation & Higher

Finding the rule

Look at the differences between terms. A constant difference means add (or subtract) that number each time; a constant ratio means multiply.

Worked Example

Find the next two terms and the rule: (a) 5, 9, 13, 17, … (b) 1, 2, 4, 8, …

Practice Question 2

Find the next term in 40, 33, 26, 19, … and state the rule.

3. The nth Term of an Arithmetic Sequence

Foundation & Higher

A linear expression

If the common difference is d, the nth term starts dn. Compare dn with the actual terms to find the adjustment. Use the nth term to decide whether a number is in the sequence: set it equal and see if n is a positive whole number.

Method nth term = (common difference) × n + (first term − common difference)

Worked Example

Find the nth term of 7, 10, 13, 16, … Is 100 a term of this sequence? Is 150?

Practice Question 3

(a) The sequence 1, 3, 5, 7, 9, … has nth term 2n − 1. Explain why. (b) The nth term of a sequence is 4n + 3. Write down the first 3 terms.

4. First Term and Common Difference

Higher tier

a, d and the nth term a + (n − 1)d

In an arithmetic sequence the first term is a and the common difference is d, so the terms are a, a + d, a + 2d, … The nth term has (n − 1) differences added.

Memorise — not on the formulae sheet nth term = a + (n − 1)d

Worked Example

The 2nd term of an arithmetic sequence is 7 and the 5th term is 19. Find a and d.

Practice Question 4

An arithmetic sequence has a = 20 and d = −3. Find the 15th term.

5. Arithmetic Series

Higher tier

Memorise

S_n

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Sum of the first n terms

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Sequences

Sequences

Term-to-term and position-to-term

Finding the rule

A linear expression

a, d and the nth term a + (n − 1)d

Sn

Past papers that test Sequences

More Mathematics A topics

S_n