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  • Find the sum to n terms of each of the series in 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 +...

1.   Find the sum to n terms of each of the series in 1 \times 2 + 2 \times 3 + 3 \times 4 + 4 \times 5 + ...

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best_answer

the series = 1 \times 2 + 2 \times 3 + 3 \times 4 + 4 \times 5 + ...

n th term  = n(n+1)=a_n

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

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

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

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

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

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

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

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

Posted by

seema garhwal

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