Question

Using mathematical induction, prove that for every positive integer n,

1 · 2 + 2 · 3 + · · · + n(n + 1) = (n(n + 1)(n + 2)) / 3 

Answer 1

Base step

for n=1: $1*2=\frac{1*2*3}{3}=2.$ The statement holds.

Inductive step

Assume that for n=k: $1*2+2*3+...+k(k+1)=\frac{k(k+1)(k+2)}{3}$

Then, for n=k+1: $1*2+2*3+...+k(k+1)+(k+1)(k+2)=\frac{k(k+1)(k+2)}{3}+(k+1)(k+2)=$

$=(k+1)(k+2)(\frac{k}{3}+1)=\frac{(k+1)(k+2)(k+3)}{3}$

That is, the statement for n=k+1 also holds true.

Since both the base step and the inductive step have been proved as true, by mathematical induction the statement holds for every natural number n.