Given that \(\mathrm { f } ( r ) = \frac { 1 } { 4 } r ^ { 2 } ( r + 1 ) ^ { 2 }\), show that
$$\mathrm { f } ( r ) - \mathrm { f } ( r - 1 ) = r ^ { 3 }$$
Use the method of differences to show that
$$\sum _ { r = n } ^ { 2 n } r ^ { 3 } = \frac { 3 } { 4 } n ^ { 2 } ( n + 1 ) ( 5 n + 1 )$$