# The nth Term

## The nth Term

The 'nth' term is a formula with 'n' in it which enables you to find any term of a sequence without having to go up from one term to the next.

'n' stands for the term number so to find the 50th term we would just substitute 50 in the formula in place of 'n'.

There are two types of sequences that you will have to deal with:

#### Constant Difference Sequences

This is when the difference between terms is always the same.

e.g. 1, 4, 7, 10, ... This has a difference which is always 3.

How do you find the formula for the 'nth' term?

Well, the three times table has the formula '3n' and the terms in this sequence are two less than the terms in the three times table so the formula is '3n - 2'.

You can always find the 'nth term' by using this formula:

nth term = dn + (a - d)

Where d is the difference between the terms, a is the first term and n is the term number.

e.g. 6, 11, 16, 21, ...For this sequence d = 5, a = 6

So the formula is nth term = 5n + (6 - 5)

which becomes nth term = 5n + 1

#### Changing Difference Sequences

What if the difference keeps changing?

Obviously these are more difficult but once again we can use a formula!

nth term = a + (n - 1)d + ½(n - 1)(n - 2)c

This time there is a letter c which stands for the second difference (or the difference between the differences and d is just the difference between the first two numbers.

Putting the right numbers into the formula is reasonably simple (once you've learnt the formula!). Simplifying it requires good Algebra skills so practice your Algebra!

Here's an example:

Here the difference between the first two numbers is 1 so d = 1

Also the second differences are 2 so c = 2 The first term is 2 so a = 2

Using the formula, nth term = 2 + (n - 1)x1 + ½(n - 1)(n - 2)x2

Getting rid of brackets (and noticing that ½ x 2 = 1):

nth term = 2 + n - 1 + n2 - 3n + 2

Simplifying,

nth term = n2 - 2n + 3

If all that gave you a headache there is an alternative way!

1. If the first differences keep changing but the second difference is constant then the formula is something to do with 'n2'. Make a table showing the first few terms of 'n2'.

2. In the next column of your table write the differences between the term of 'n2' and your sequence.

3. Find the formula for this new sequence using dn + (a - d)

4. Add it on to n2 to give yo your final formula.

Have a look at this using the sequence above:

 Sequence 'n2' Difference 2 1 1 3 4 -1 6 9 -3 11 16 -5 18 25 -7

For the sequence 1, -1, -3, -5, -7 => a = 1 and d = -2

So the formula is -2n + (1 - -2) which simplifies to -2n + 3

Final Formula (Step 4) : nth term = n2 -2n + 3 (as we got before!)

Try these out using both methods and decide which one you prefer.

#### Test yourself

Drag and drop the solutions into the right spaces, then click on 'mark answer' to see if you are right: