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If Sn=1+3+6+10+...+n(n+1)/2 then Sn is

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HEY FRIEND, HARISH here.


Here is your ANSWER.


\mathrm{S_n = 1+3+6+...\frac{n(n+1)}{2}} \\ \\ \sum\limits_{k=1}^n\Bigl( \frac{k(k+1)}{2} \Bigr) = \frac{1}{2} \sum\limits_{k=1}^n\Bigl( k^2 + k \Bigr ) \\ \\ \\ \implies \frac{1}{2} \sum\limits_{k=1}^n\Bigl( k^2 \Bigr ) + \sum\limits_{k=1}^n\Bigl( k \Bigr) \\ \\ \\ \implies \frac{1}{2} \Bigl( \frac{n\left(n+1\right)\left(2n+1\right)}{6}+\frac{n\left(n+1\right)}{2} \Bigr) \\ \\ \\ \implies \frac{1}{2} \Bigl( \frac{n\left(n+1\right)\left(n+2\right)}{3} \Bigr) \\ \\ \\ \boxed{\bold{\frac{n(n+1)(n+2)}{6}}}

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