If 1³ + 2³ . . . . . . + 10³ = 3025, then the value of 2³ + 4³ . . . . . . + 20³ is:

Solution (By Examveda Team)

$$\eqalign{ & \because {1^3} + {2^3} + {3^3}\,......\, + {10^3} = 3025 \cr & {\text{To find}} = {2^3} + {4^3} + {6^3}\,......\, + {20^3} = ? \cr & \Rightarrow {2^3}\left( {{1^3} + {2^3} + {3^3}\,......\, + {{10}^3}} \right) \cr & \left[ {{{\left( {\frac{{n\left( {n + 1} \right)}}{2}} \right)}^2} = {{\left( {\frac{{10\left( {11} \right)}}{2}} \right)}^2} = {{\left( {55} \right)}^2} = 3025} \right] \cr & \Rightarrow 8 \times 3025 \cr & \Rightarrow 24200 \cr} $$