Question

Find the sum of the series ${31}^3+{32}^3+...+{50}^3.$

Answer 1

$S=(1^3+2^3+....+{50}^3)-(1^3+2^3+....+{30}^3)$

$={\left(\frac{50\times 51}{2}\right)}^2-{\left(\frac{30\times 31}{2}\right)}^2$

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

$=\frac{1}{4}(50\times 51-30\times 31)(50\times 51+30\times 31)$

$=1409400.$