Question

Is $\mathrm{1.2.3}+\mathrm{2.3.4}+\mathrm{3.4.5}+...n(n+1)(n+2)$=$\frac{n(n+1)(n+2)(n+3)}{4}$ true or false??

• A. True
• B. False

Answer 1

The correct option is A True

$\mathrm{1.2.3}+\mathrm{2.3.4}+.....+n(n+1)(n+2)=\sum n(n+1)(n+2)$

$=\sum n(n^2+3n+2)$

Applying $\sum$ to every number

$=\sum n^3+\sum 3n^2+\sum 2n$

$=\frac{n^2(n+1{)}^2}{4}+\frac{3\left[n\right(n+1\left)\right(2n+1\left)\right]}{6}+\frac{2n(n+1)}{2}$

$=\frac{n(n+1)}{2}\{\frac{n(n+1)}{2}+(2n+1)+2\}$

$=\frac{n(n+1)}{2}\frac{(n^2+n)+(4n+2)+4}{2}$

$=\frac{n(n+1)(n²+5n+6)}{4}$

$=\frac{n(n+1)(n+2)(n+3)}{4}$