In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
Explain why, for \(n \in \mathbb { N }\)
$$\sum _ { r = 1 } ^ { 2 n } ( - 1 ) ^ { r } \mathrm { f } ( r ) = \sum _ { r = 1 } ^ { n } ( \mathrm { f } ( 2 r ) - \mathrm { f } ( 2 r - 1 ) )$$
for any function \(\mathrm { f } ( r )\).
Use the standard summation formulae to show that, for \(n \in \mathbb { N }\)
$$\sum _ { r = 1 } ^ { 2 n } r \left( ( - 1 ) ^ { r } + 2 r \right) ^ { 2 } = n ( 2 n + 1 ) \left( 8 n ^ { 2 } + 4 n + 5 \right)$$
Hence evaluate
$$\sum _ { r = 14 } ^ { 50 } r \left( ( - 1 ) ^ { r } + 2 r \right) ^ { 2 }$$