Cube of 20 is Sum of Sequence of 4 Consecutive Cubes

Theorem

$20^3 = \ds \sum_{k \mathop = 11}^{14} k^3$

That is:

$20^3 = 11^3 + 12^3 + 13^3 + 14^3$

$\ds \sum_{k \mathop = 1}^{14} k^3$

$=$

$\ds \paren {\dfrac {14 \paren {14 + 1} } 2}^2$

$\ds $

$=$

$\ds 11 \, 025$

$\ds \sum_{k \mathop = 1}^{10} k^3$

$=$

$\ds \paren {\dfrac {10 \paren {10 + 1} } 2}^2$

$\ds $

$=$

$\ds 3025$

$\ds \leadsto \ \ $

$\ds \sum_{k \mathop = 11}^{14} k^3$

$=$

$\ds \sum_{k \mathop = 1}^{14} k^3 - \sum_{k \mathop = 1}^{10} k^3$

$\ds $

$=$

$\ds 11 \, 025 - 3025$

$\ds $

$=$

$\ds 8000$

$\ds $

$=$

$\ds 20^3$

$\blacksquare$