Question

Prove that for every positive integer n,

$\begin{array}{r}\text{1.2.}+2.3.+\dots +n(n+1)\\ =n(n+1)(n+2)/3\end{array}$

Short answer

$\begin{array}{r}\text{1.2.}+2.3.+\dots +n(n+1)\\ =n(n+1)(n+2)/3\end{array}$

Step-by-step solution

Step: 1

Basic step: so, for n=1,

$\begin{array}{r}P\left(1\right)=1.2=\frac{1(1+1)(1+2)}{3}\\ 2=2\end{array}$

True for n=1.

Step: 2

Assume for n=k,

$P\left(k\right)=1.2\dots k(k+1)=\frac{k(k+1)(k+2)}{3}$

Step: 3

To prove for n=k+1

$\begin{array}{r}P\left(k\right)=1.2\dots k(k+1)+(k+1)(k+2)\\ =\frac{k(k+1)(k+2)}{3}+(k+1)(k+2)\\ =\frac{(k+1)(k+2)(k+3)}{3}\\ =\frac{(k+1)(k+1)+1(k+1)+2}{3}\end{array}$

True for n=k+1.