Question

Solve $S={15}^3+{16}^3+{17}^3+......+{60}^3$

Answer 1

$S={15}^3+{16}^3+....+{60}^3$

$S=1^3+2^3+.....+{60}^3-\left(1^3+2^3+...+{14}^3\right)$

$\sum n^3={\left[\frac{n\left(n+1\right)}{2}\right]}^2$

$S={\left[\frac{60-\left(60+1\right)}{2}\right]}^3-{\left[\frac{14\left(14+1\right)}{2}\right]}^2$

${\left(30\times 61\right)}^3-{\left(7\times 15\right)}^3$

${\left(1830\right)}^3-{\left(105\right)}^3$

$6128487000-1157625$

$6127329375$