Revision:Sequences

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Introduction

In the sequence 2, 4, 6, 8, 10... there is an obvious pattern. Such sequences can be expressed in terms of the n^th term of the sequence. In this case, the n^th term 2n. To find the 1^st term, put n = 1 into the formula, to find the 4^th term, replace the n's by 4's: 4^th term 2 × 4 = 8.

Example

What is the n^th term of the sequence 2, 5, 10, 17, 26... ? The easiest way to find the nth term is by trial and error.

n | 1 - 2 - 3 - 4 - 5

n² | 1 - 4 - 9 - 16 - 25

n² + 1 | 2 - 5 - 10 - 17 - 26

This is the required sequence, so the nth term is n² + 1. For some sequences, there is no easy way of working out the n^th term of a sequence, other than to try different possibilities.

Tips: if the sequence is going up in threes (e.g. 3, 6, 9, 12...), there will probably be a three in the formula, etc.

In many cases, square numbers will come up, so try squaring n, as above. Also, the triangular numbers formula often comes up. This is .

Notation

The n^th term of a sequence is usually written as Un . So in the last example, U_n = n² + 1 . The 5^th term is therefore U₅ = 25 + 1 26.

Recurrence relation

This is where the next term of a sequence is defined using the previous term(s). For example, the recurrence relation for 2, 4, 8, 16, 32, ... would be: U₁ = 2, U_n = 2(U_n-1). This tells us that the first term, U₁, is 2 and the next term of the sequence can be found by doubling the previous term.

Convergent Sequences

Sequences whose nth term approaches a finite number as n becomes larger are known as convergent sequences and the number to which the sequence converges is known as the limit of the sequence. For example: 10, 5, 2.5, 1.25, 0.625, ... converges towards the limit zero.

Comments

These notes are missing 'simpler' info needed at GCSE (generating sequences, more details on finding nth terms etc) while are also missing the more advanced topics needed for A Level.