6. (a) Prove by induction $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$ (b) Using the result in part (a), show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 3 } - 2 \right) = \frac { 1 } { 4 } n \left( n ^ { 3 } + 2 n ^ { 2 } + n - 8 \right)$$ (c) Calculate the exact value of \(\sum _ { r = 20 } ^ { 50 } \left( r ^ { 3 } - 2 \right)\).