Question about Un = Sn - S(n-1) formula (Sequences and series)

Hello, I was doing a paper from the Madasmaths collection (FP1 Paper O), and I got confused when looking at the solution of question 8 part b (see attached files). Basically, the answer seems to assume that in order to find the nth term, Un, of the sequence that form the series, we just find the last term. So, using the formula Un = Sn S(n-1) we manage with some manipulation to find the requested expression. The problem is that the series Sn = n^{2}(n+1)(n+2) is not an arithmetic series. If we sub 1, 2, 3, into n, we get 6, 48, 180, 480, …, which doesn’t seem arithmetic to me. My understanding being that the last term is the nth term only in arithmetic sequences, I can’t see how the assumption on which the answer rests can hold…

Can anyone help ?

Thank you
Sn is the sum of a sequence u1, u2, ... , un. So defining S1=u1, then
Sn = S_(n-1) + un
is a recursive way of expressing it. As Sn is a quartic in n, youd expect un to be a cubic which is what they found.
(edited 1 month ago)
Thank you