bantyr math

He has made a small mistake in a sign.

It is evident that the sequence {S_i+1-S_i} is an AP.

So, that means S_n+3-S_n+2, S_n+2-S_n+1, S_n+1-S_n are in A.P.

(S_n+3-S_n+2)-(S_n+2-S_n+1) =(S_n+2-S_n+1) - (S_n+1-S_n)

Rearranging the terms gives S_n+3-3S_n+2+3S_n+1-S_n=0

In the theory of finite differences LHS is known as the 3rd order forward difference operator, and that it equals zero tells us that the sequence is characterised by a quadratic, i.e. t_n =an²+bn+c