Question

A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely. Show your work.

Sn: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3

Answer 1

$\displaystyle S_k = \sum\limits_{i=1}^k{i(i+1)} = \dfrac{k(k+1)(k+2)}{3} \\ \\ S_{k+1} = \sum\limits_{i=1}^{k+1}{i(i+1)} = \dfrac{(k+1)(k+1+1)(k+1+2)}{3}=\\ \\ = \dfrac{(k+1)(k+2)(k+3)}{3}$