3.1 Sequences

REVISION NOTES

IGCSE Edexcel Mathematics A

3.1.1 Generate terms of a sequence using term-to-term and position-to-term definitions of the sequence

A sequence is an systematic set of (most of the time) numbers
Each of the numbers in the sequence is called a term
The location of a term within a sequence is called its position. (eg. n is an unknown position
For the first term, n=1 and for the second term, n=2, and so on
Subscript notation is used to talk about a particular term
- a1 would be the first term in a sequence
- a7 would be the seventh term
- an would be the nth term
A position-to-term rule gives the nth term of a sequence in terms of n
- This is a very powerful piece of mathematics
- With a position-to-term rule the 100th term of a sequence can be found without having to know or work out the first 99 terms!
A term-to-term rule gives the (n+1)th term in terms of the nth term
- ie an+1 is given in terms of an
- If a term is known, the next one can be worked out

How do I use a position-to-term rule to write the first n terms of a sequence?

To generate the first n terms of a sequence using a position-to-term rule (an nth term rule), substitute n = 1, n = 2, n = 3, and so on, into the rule
For example, using an = 2n + 3 to generate the first four terms;
- 1st term: n = 1 so a1 = 2(1) + 3 = 5
- 2nd term: n = 2 so a2 = 2(2) + 3 = 7
- 3rd term: n = 3 so a3 = 2(3) + 3 = 9
- 4th term: n = 4 so a4 = 2(4) + 3 = 11
- Sequence is 5, 7, 9, 11, …
Another example, using an = n2 − 5 to generate the first four terms;
- a1 = 12 − 5 = -4
- a2 = 22 − 5 = -1
- a3 = 32 − 5 = 4
- a4 = 42 − 5 = 11
- Sequence is -4, -1, 4, 11, … (This is an example of a quadratic sequence)

How do I use a term-to-term rule to write the first n terms of a sequence?

To generate the first n terms of a sequence using a term-to-term rule (an nth term rule), you need to be given the first term (a1) and the term-to-term rule
The term-to-term rule may be given in the form “an+1 = … ” where an+1 means the next term
For example, a1 = 5 and an+1 = an + 2, generate the first four terms;
- 1st term, a1 = 5
- 2nd term, a2 = a1 + 2 = 5 + 2 = 7
- 3rd term, a3 = a2 + 2 = 7 + 2 = 9
- 4th term, a4 = a3 + 2 = 9 + 2 = 11
- Sequence is 5, 7, 9, 11, … Notice that this is the same sequence that was generated above using the position-to-term rule an = 2n + 3
Another example, a1 = 5 and an+1 = 2an, generate the first four terms;
- a1 = 5
- a2 = 2a1 = 2 × 5 = 10
- a3 = 2a2 = 2 × 10 = 20
- a4 = 2a4 = 2 × 20 = 40
- Sequence is 5, 10, 20, 40, … (This is an example of a geometric sequence)
A final example, a1 = 1, a2 = 1 and an+2 = an+1 + an, generate the first four terms;
- a1 = 1 and a2 = 1
- a3 = a2 + a1 = 1 + 1 = 2
- a4 = a3 + a2 = 2 + 1 = 3
- a5 = a4 + a3 = 3 + 2 = 5
- Sequence is 1, 1, 2, 3, 5, … (This is the Fibonacci sequence)