Question

How do you write the expression for the nth term of the sequence given `0, 3, 8, 15, 24,...`?

Answer 1

`n^2-1`

Explanation:

If you try to find the differences between the numbers of the sequence, you'll notice that they go up by 3, 5, 7 and 9. These differences have a second difference of 2(i.e. they go up by 2).

Remember that the structure of a quadratic sequence, is `an^2±bn±c`

So, you halve the second difference in order to get the `a` coefficient, which is 1, to get:
`n^2±bn±c`

Now, if you subtract the original sequence by the sequence of `n^2`, you'll notice that there is a common difference, -1. Therefore, there is no additional sequence that you have to take into consideration.

Thus, the answer is: `n^2-1`!

(Also, if you know your squared numbers well, you will notice that it is `n^2-1` without having to do much calculation :D)

Hope this helps!