1 , + , 1^3 , + , 2^3 1 + 2 + 1^3 , + , 2^3 + 3^3 1 + 2 + 3 + ...… - Vidyadip

MCQ

$1\, + \,\frac{{{1^3}\, + \,{2^3}}}{{1 + 2}} + \frac{{{1^3}\, + \,{2^3} + {3^3}}}{{1 + 2 + 3}} + ...... + \frac{{{1^3}\, + \,{2^3} + {3^3} + ..... + {{15}^3}}}{{1 + 2 + 3 + ..... + 15}} - \frac{1}{2}\left( {1 + 2 + 3 + ....+15} \right)$ is equal to

✓
$620$
B
$1860$
C
$1240$
D
$660$

✓

Answer

Correct option: A.

$620$

a
Sum $ = \sum\limits_{n = 1}^{15} {\frac{{{1^3} + {2^3} + ...{n^3}}}{{1 + 2 + ... + n}}} - \frac{1}{2}.\frac{{15.16}}{2}$

$ = \sum\limits_{n = 1}^{15} {\frac{{n\left( {n + 1} \right)}}{2} - 60} $

$ = \sum\limits_{n = 1}^{15} {\frac{{n\left( {n + 1} \right)\left( {n + 2 - \left( {n - 1} \right)} \right)}}{6}} - 60$

$ = \frac{{15.16.17}}{6} - 60 = 620$

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