Struggling with the very first exercise in Polya's "Mathematics and plausible reasoning" Find and proove general law.
1 = 0 + 1
2 + 3 + 4 = 1 + 8
5 + 6 + 7 + 8 + 9 = 8 + 27
10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64
I looked at the right part and found out $$ a_n = (n -1)^3 + (n)^3 $$ tried to use induction to prove, but no way expanded formula of $$a_n = 2n^3 + 3n^2 + 3n + 1 $$ would equivalent to $$a_{n+1} = 2n^3 - 3n^2 + 3n - 1 $$ so this was all I was capable of. Don't understand how could I get the right answer and what's going on the left side of it.
The left-hand side $$(n^2+1)+(n^2+2)+\dots+(n+1)^2$$ can be written as $$(1+2+\dots+(n+1)^2)-(1+2+\dots+n^2).$$ Now you probably know the formula $1+2+\dots+k=\frac{k(k+1)}{2}$. Just apply this formula to each of the two sums.