4. (a) Show that $$r ^ { 2 } ( r + 1 ) ^ { 2 } - ( r - 1 ) ^ { 2 } r ^ { 2 } \equiv 4 r ^ { 3 }$$ Given that \(\sum _ { r = 1 } ^ { n } r = \frac { 1 } { 2 } n ( n + 1 )\)
(b) use the identity in (a) and the method of differences to show that $$\left( 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } + \ldots + n ^ { 3 } \right) = ( 1 + 2 + 3 + \ldots + n ) ^ { 2 }$$