Question

find the sum of 1×2+2×3+3×4+......upto n(n+1) terms​

Answer 1

Step-by-step explanation:

We have,

$1 \times 2 + 2 \times 3 + 3 \times 4 + .... + n(n + 1)$

$= \sum^{n}_{k = 1}k(k + 1)$

$= \sum^{n}_{k = 1} ({k}^{2} + k )$

$= \sum^{n}_{k = 1} {k}^{2} + \sum^{n}_{k = 1}k$

$= \frac{n(n + 1)(2n + 1)}{6} + \frac{n(n + 1)}{2}$

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

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

$= \frac{2n(n + 1)(n + 2)}{6}$

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