Standard +0.3 This is a straightforward Further Maths question requiring algebraic manipulation of standard summation formulas (expanding r²(r+1) into r³+r²) and then using the difference method for part (b). While it involves multiple steps, the techniques are routine for F1 students with no novel insight required, making it slightly easier than average.
6. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 1 ) = \frac { n } { a } ( n + 1 ) ( n + 2 ) ( 3 n + b )$$
where \(a\) and \(b\) are integers to be found.
(b) Hence find the value of
$$\sum _ { r = 25 } ^ { 49 } \left( r ^ { 2 } ( r + 1 ) + 2 \right)$$
6. (a) Use the standard results for $\sum _ { r = 1 } ^ { n } r ^ { 2 }$ and for $\sum _ { r = 1 } ^ { n } r ^ { 3 }$ to show that, for all positive integers $n$,
$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 1 ) = \frac { n } { a } ( n + 1 ) ( n + 2 ) ( 3 n + b )$$
where $a$ and $b$ are integers to be found.\\
(b) Hence find the value of
$$\sum _ { r = 25 } ^ { 49 } \left( r ^ { 2 } ( r + 1 ) + 2 \right)$$
\hfill \mbox{\textit{Edexcel F1 2017 Q6 [8]}}