Standard +0.8 This is a Further Maths question requiring expansion of a cubic product, manipulation of standard summation formulas, and algebraic simplification to match a given form. Part (b) adds the non-trivial step of computing a sum from r=10 to 20 using the difference of sums. While systematic, it requires careful algebra across multiple steps and is more demanding than typical A-level questions.
4. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } r ( 2 r + 1 ) ( 3 r + 1 ) = \frac { 1 } { 6 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$
where \(a\), \(b\) and \(c\) are integers to be determined.
(b) Hence find the value of
$$\sum _ { r = 10 } ^ { 20 } r ( 2 r + 1 ) ( 3 r + 1 )$$
4. (a) Use the standard results for $\sum _ { r = 1 } ^ { n } r , \sum _ { r = 1 } ^ { n } r ^ { 2 }$ and $\sum _ { r = 1 } ^ { n } r ^ { 3 }$ to show that, for all positive integers $n$,
$$\sum _ { r = 1 } ^ { n } r ( 2 r + 1 ) ( 3 r + 1 ) = \frac { 1 } { 6 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$
where $a$, $b$ and $c$ are integers to be determined.\\
(b) Hence find the value of
$$\sum _ { r = 10 } ^ { 20 } r ( 2 r + 1 ) ( 3 r + 1 )$$
\hfill \mbox{\textit{Edexcel F1 2017 Q4 [7]}}