Proof of Sequences: Orders and Representations

[courtrigrad]

Hello all

Let us say we are given a sequence of order 2. By order 2 I mean that we have a sequence in which the differences between the terms forms a sequence of order 1, which has a constant difference between terms. How can I prove that the nth term of a sequence of order 2 can be represented as:

an^2 + bn + c?


Or more generally how would I prove that that the nth term of a sequence of order k can be represented as:

an^k + bn^k-1 +.... + pn + q?

Any help would we greatly appreciated.

Thanks

[Tide]

I think you only need to focus on the highest order term. Just look at the difference P(n+1) - P(n) where P(n) is your sequence and use the first term in the binomial expansion of P(n+1).