Standard +0.8 This Further Maths question requires applying standard summation formulas, algebraic manipulation to match a given form and find constants a and b, then using partial sums with an additional term involving k. While the techniques are standard for F1, the multi-step nature, algebraic complexity of matching the factored form, and the need to handle the r² term in part (b) elevate this above typical A-level questions.
5. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } - 3 r \right) = \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 3 ) ( a n + b )$$
where \(a\) and \(b\) are integers to be found.
Given that
$$\sum _ { r = 5 } ^ { 10 } \left( 8 r ^ { 3 } - 3 r + k r ^ { 2 } \right) = 22768$$
(b) find the exact value of the constant \(k\).
5. (a) Use the standard results for $\sum _ { r = 1 } ^ { n } r$ and $\sum _ { r = 1 } ^ { n } r ^ { 3 }$ to show that, for all positive integers $n$,
$$\sum _ { r = 1 } ^ { n } \left( 8 r ^ { 3 } - 3 r \right) = \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 3 ) ( a n + b )$$
where $a$ and $b$ are integers to be found.
Given that
$$\sum _ { r = 5 } ^ { 10 } \left( 8 r ^ { 3 } - 3 r + k r ^ { 2 } \right) = 22768$$
(b) find the exact value of the constant $k$.
\hfill \mbox{\textit{Edexcel F1 2016 Q5 [8]}}