- (a) Use the standard summation formulae to show that, for \(n \in \mathbb { N }\),
$$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 17 r - 25 \right) = n \left( n ^ { 2 } - A n - B \right)$$
where \(A\) and \(B\) are integers to be determined.
(b) Explain why, for \(k \in \mathbb { N }\),
$$\sum _ { r = 1 } ^ { 3 k } r \tan ( 60 r ) ^ { \circ } = - k \sqrt { 3 }$$
Using the results from part (a) and part (b) and showing all your working,
(c) determine any value of \(n\) that satisfies
$$\sum _ { r = 5 } ^ { n } \left( 3 r ^ { 2 } - 17 r - 25 \right) = 15 \left[ \sum _ { r = 6 } ^ { 3 n } r \tan ( 60 r ) ^ { \circ } \right] ^ { 2 }$$