Find the sequence of differences for the given sequence. next, find a closed form solution for the following sequence. (be sure to refer to the example on the modules notes page for “closed form solutions.”)
1, 5, 11, 19, 29, 41, …

Let denote the given sequence with :

Let be the sequence of the forward differences of , so that for :

follows an arithmetic progression with a difference of 2 between terms, so that

Then we have

so that is given recursively by

By substitution, we can try to find a pattern:

and so on, with the general pattern

and since we can write this as

Recall that

Then