The sum [1 + 1^3 + 2^3 1 + 2 + 1^3 + 2^3 + 3^3 1 + 2 + 3 + ......

Question

The sum

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

Accepted Answer

$$Sum = \sum\limits_{n = 1}^{15} {{{{1^3} + {2^3} + .... + {n^3}} \over {1 + 2 + .... + n}}}  - {1 \over 2}{{15 \times 16} \over 2}$$

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

$$ = {1 \over 2}\sum\limits_{n = 1}^{15} {{n^2}}  + {1 \over 2}\sum\limits_{n = 1}^{15} {n - 60} $$

$$ = {1 \over 2} \times {{15 \times 16 \times 31} \over 6} + {1 \over 2} \times {{15 \times 16} \over 2} - 60$$

= 620

Solutions · 1