Given that \(\mathrm { f } ( r ) = r ^ { 2 } \left( 2 r ^ { 2 } - 1 \right)\), show that
$$\mathrm { f } ( r ) - \mathrm { f } ( r - 1 ) = ( 2 r - 1 ) ^ { 3 }$$
Use the method of differences to show that
$$\sum _ { r = n + 1 } ^ { 2 n } ( 2 r - 1 ) ^ { 3 } = 3 n ^ { 2 } \left( 10 n ^ { 2 } - 1 \right)$$
(4 marks)