Learning Goal

Syllabus Requirement

Students need to learn aboutsequences, with the below point being highlighted in particular:

•    Sequences given by a formula for the ??n^{th}?? term; and
•    Sequences generated by a simple relation of the form ??x_{n+1}=f(x_{n})??.

Study Notes

A sequences is a series of numbers following a set rule, such as ??2, \: 4, \:, 6, \:, 8\: ...??. Each number or expression in a sequence is called a term.

The notation ??U_{n}?? is often used to refer to the ??n^{th}?? term of a sequence. So ??U_{1}?? is the first term, ??U_{2}?? the second term and so on.

There are two ways in which a general sequence is usually specified:

The 2 ways of specifying a general sequence.

1. Formula for the ??n^{th}?? term.
If a formula for the ??n^{th}?? term is given, then any term in the sequence can be found by substituting the position of the term you require into the formula.

2. Using a simple relationship between terms (also known as a 'recurrence relation').
A recurrence relation can define a sequence, often by giving the first term of the sequence and a relationship that links any term to the previous term.

Worked Examples

We'll now run through some examples of each of the two ways of specifying a general sequence.

1. Formula for the ??n^{th}?? term.

Example 1

Question

The ??n^{th}?? term of a sequence is given by ??u_{n} = 3n + 1??. What is the ??10^{th}?? term?

Example 2

Question

The ??n^{th}?? term of a sequence is given by ??u_{n} = \frac{n^{2}}{n+1}??. What is the ??3^{rd}?? term?

Example 3

Question

Find the value of ??n??, where the ??n^{th}?? term is given by ??2n - 9??, where ??U_{n} = 13??.

2. Using a simple relationship between terms (also known as a 'recurrence relation').

Example 4

Question

If ??U_{1} = 3?? and the recurrence relation is ??U_{n+1} = 3U_{n} - 7??, find the first ??3?? terms of the sequence.

Example 5

Question

If ??U_{1} = 5?? and ??U_{2} = 8??, find ??m?? in the recurrence relation ??U_{n+1} = mU_{n} - 2??.

Exam Questions

The table below contains every exam question that has been asked on this topic, this includes normal papers from both January and June sittings, International papers and Specimen papers.

To see an exam question and solution simply click on the load question icon (), the question will appear below the table and the solution can be shown by clicking the "Show Solution" button that also appears.

The full exam paper and mark schemes are also available for download by clicking on the download icons in each row of the table ().

Exam Board Subject Paper Year Month Module Question
No.
Parts Total
Marks
Question
Exam
Paper
Mark
Scheme
Edexcel Maths Standard 2006 January Core 1 2 (a), (b) 4
Edexcel Maths Standard 2006 June Core 1 4 (a), (b) 5
Edexcel Maths Standard 2007 June Core 1 8 (a), (b), (c) 7
Edexcel Maths Standard 2008 January Core 1 7 (a), (b), (c), (d) 8
Edexcel Maths Standard 2008 June Core 1 5 (a), (b), (c) 6
Edexcel Maths Standard 2009 June Core 1 7 (a), (b), (c) 7
Edexcel Maths Standard 2010 June Core 1 5 (a), (b) 4
Edexcel Maths Standard 2011 January Core 1 4 (a), (b) 5
Edexcel Maths Standard 2011 June Core 1 5 (a), (b), (c) 7
Edexcel Maths Standard 2012 January Core 1 4 (a), (b), (c) 6
Edexcel Maths Standard 2012 June Core 1 5 (a), (b), (c) 7
Edexcel Maths Standard 2013 January Core 1 4 (a), (b) 5
Edexcel Maths Standard 2013 June Core 1 4 (a), (b) 7
Edexcel Maths International 2013 June Core 1 6 (a), (b), (c), (d) 9

Exam Tips

1. Exam questions will often give you a recurrence relation and ask you to find values later in the sequence, this can be done by substituting into the relation given.

2. Sometimes exam questions will ask you to find the range of possible values of a constant in the recurrence relation, typically you'll need to set the expression for a term or sum of terms to some value, which will allow you to solve for the range of possible values of the constant.

3. You should be familiar with sum notation, as this is often used in questions. Remember, there is no calculator allowed, so you will be typically only asked to add up a handful of terms. If you are asked to find the sum to a larger number of terms, say ??100??, then you will be expected to spot some pattern or trend to help you do this - do not try to add up an unreasonable number of terms by hand!