$(n-1)^2 + (n-1) + 1 = n^2 - n + 1$ telescope

$good hint$

This is a telescoping function Expanding f(n) Lets use n = 1 : n^2 + n + 1 (n + 1)^2 + (n + 1) + 1 = n^2 + 3n + 3 (n + 2)^2 + (n + 2) + 1 = n^2 + 5n + 7 (n + 3)^2 + (n + 3) + 1 = n^2 + 7n + 13 … Expanding g(n) n^2 - n + 1 (n + 1)^2 - (n + 1) + 1 = n^2 + n + 1 (n + 2)^2 - (n + 2) + 1 = n^2 + 3n + 3 (n + 3)^2 - (n + 3) + 1 = n^2 + 5n + 7 Notice that there are repeated terms Product sum f(n) from 1 to 100 : Product sum g(n) from 1 to 100 100^2 + 100 + 1 : 1^2 - 1 + 1 10101 : 1