Find the next three terms in this sequence
14, 24, 36, 50, 66, ..., ..., ...,
As with any sequence we are given first let's look to see if it is not one we know. It's not one I know, so let's find the first difference between adjacent terms.
This first difference is an arithmetic sequence with a common difference of 2, which enables us to work out the next three first differences:
Adding these first differences in term to the previous term in the sequence we get:
So the sequence is 14, 24, 36, 50, 66, 84, 104, 126,…
We can go further and find any term in the sequence. The first difference is an arithmetic sequence with a common difference of 2, implying that the original sequence is a quadratic sequence of the form , where a is the common difference divided by 2, ∴ .
But , so the last line is an arithmetic sequence with common difference and zeroth term . Therefore, the term of the sequence is given by .