Prove the following

Question:

The sum $1+\frac{1^{3}+2^{3}}{1+2}+\frac{1^{3}+2^{3}+3^{3}}{1+2+3}+\ldots$

$\left.+\frac{1^{3}+2^{3}+3^{3}+\ldots .+15^{3}}{1+2+3+\ldots .+15}-\frac{1}{2}(1+2+3+\ldots .+15)\right]$

  1. 1240

  2. 1860

  3. 660

  4. 620


Correct Option: , 4

Solution:

$\operatorname{Sum}=\sum_{n=1}^{15} \frac{1^{3}+2^{3}+\ldots n^{3}}{1+2+\ldots .+n}-\frac{1}{2} \cdot \frac{15.16}{2}$

$=\sum_{n=1}^{15} \frac{n(n+1)}{2}-60$

$=\sum_{n=1}^{15} \frac{n(n+1)(n+2-(n-1))}{6}-60$

$=\frac{15.16 .17}{6}-60=620$

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