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  4. The sum 1+ (13 +23/1+2) + (13+23+33/1+2+3) +.... + (13 +23+33 +....+153/1+2+3+...+15) - (1/2) (1+2+3+...+15)

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Q. The sum $1+ \frac{1^{3} +2^{3}}{1+2} + \frac{1^{3}+2^{3}+3^{3}}{1+2+3} +.... + \frac{1^{3} +2^{3}+3^{3} +....+15^{3}}{1+2+3+...+15} - \frac{1}{2} \left(1+2+3+...+15\right)$

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Solution:

Sum $= \sum^{15}_{n=1} \frac{1^{3} + 2^{3} +...n^{3}}{1+2+...+n} - \frac{1}{2}. \frac{15.16}{2} $
$ =\sum^{15}_{n=1} \frac{n\left(n+1\right)}{2} -60 =\sum^{15}_{n=1} \frac{n\left(n+1\right)\left(n+2-\left(n-1\right)\right)}{6} -60 $
$ = \frac{15.16.17}{6} - 60 = 620 $