Hi,

Suppose I give you the sequence of numbers

1/1, 1/4, 1/9, 1/16, ...

The ... indicates that the pattern is to continue. The pattern I see is that these terms are all 1 divided by a square,

1/1², 1/2², 1/3², 1/4², ...

I want an expression that tells you what every term is. I know the terms are 1 over some quantity squared and I am going to call this quantity n. Thus the terms are all 1/n² for some number n. Now I can see what n is, it is the term number

The first term is 1/1²,
the second term is 1/2²,
the third term is 1/3²,
so the nth term is 1/n².

Let's look at a different example

3, 7, 11, 15, ...

Here what I see is that each term is 4 more than the previous term, so I can write the sequence as

3, 3 + (1 4), 3 + (2 4), 3 + (3 4), ...

In this case every term after the first is 3 plus some quantity times 4, and this quantity is 1 or 2 or 3 or ... I can make this pattern work, even for the first term if I write the sequence

3 + (0 4), 3 + (1 4), 3 + (2 4), 3 + (3 4), ...

Thus

the first term is 3 + (0 4)
the second term is 3 + (1 4)
the third term is 3 + (2 4)

In this case the quantity multiplied by 4 is one less than the term number so

the nth term is 3 + ((n-1) 4)

I hope this helps,
Penny