FINDING THE Nth TERM - RGS Info



Finding the nth Term

This is also known as finding a formula for the sequence of numbers.

Consider the sequence of numbers

1 4 9 16 25 36…

Hopefully we recognise these as the square numbers

12 22 32 42 52 62….

So we could say that the general term was n2

We write

Un = n2

[pic]

Un is used as shorthand for “the nth term”

Some sequences are not so easy to spot though!

Consider the sequence

4 7 10 13 16…..

In this sequence there is a common difference between the terms of 3 so the formula must contain 3n.

The term before the sequence starts in this case is 1

So the formula becomes

Un = 3n + 1

Similarly the sequence

3 8 13 18 23 28…..

would become

Un = 5n – 2

And if the sequence decreases

21 17 13 9 5….

would become

Un = -4n + 25

or better written as

Un = 25 - 4n

If you have an nth term formula, you can use it to find the value of any term quickly:

Example 1

An arithmetic sequence starts 30 34 38 42 …, what is the 100th term?

30 34 38 42 … has a common difference of 4

The previous term would be 26

So, Un = 4n +26

So, U100 = 4 × 100 +26 = 426

This is much quicker than listing the whole sequence!

Example 2

An arithmetic sequence starts 48 41 34 27 …, what is the 100th term?

48 41 34 27 … has a common difference of -7

The previous term would be 55

So, Un = -7n + 55 = 55 – 7n

So, U100 = 55 – 7 × 100 = -645

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