Question

Prove that $1.2+2.3+.......n(n+1)$=$\frac{n(n+1)(n+2)}{3}$ in mathematical induction.

Answer 1

We shall prove the result by principle of mathematical induction.

checking for $n=1$,

$LHS:1.2=2$

$RHS:\frac{1}{3}\times 1\times 2\times 3=2$.

Hence true for $n=1$

Let us assume the result is true for $n=k$ ie.,

$1.2+2.3+.....k(k+1)=\frac{1}{3}\times k\times (k+1)\times (k+2)$

We shall prove the result to be true for $n=k+1$.

that is, to prove $1.2+2.3.....+k(k+1)+(k+1)(k+2)=\frac{1}{3}(k+1)(k+2)(k+3)$

consider $LHS:1.2+2.3.....+k(k+1)+(k+1)(k+2)$

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

$=(k+1)(k+2)\left[\frac{1}{3}\right(k+1\left)\right]$

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

$=RHS$.

Hence the result holds for $n=k+1$.

Hence proof is complete by PMI and therefore the result holds.