Standard +0.3 This is a straightforward Further Maths question requiring standard summation formulae. Part (a) is routine algebraic manipulation of given results, while part (b) adds a geometric series calculation. Both parts follow well-established procedures with no novel problem-solving required, making it slightly easier than average even for Further Maths.
5. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that
$$\sum _ { r = 1 } ^ { n } \left( 9 r ^ { 2 } - 4 r \right) = \frac { 1 } { 2 } n ( n + 1 ) ( 6 n - 1 )$$
for all positive integers \(n\).
Given that
$$\sum _ { r = 1 } ^ { 12 } \left( 9 r ^ { 2 } - 4 r + k \left( 2 ^ { r } \right) \right) = 6630$$
(b) find the exact value of the constant \(k\).
5. (a) Use the standard results for $\sum _ { r = 1 } ^ { n } r$ and $\sum _ { r = 1 } ^ { n } r ^ { 2 }$ to show that
$$\sum _ { r = 1 } ^ { n } \left( 9 r ^ { 2 } - 4 r \right) = \frac { 1 } { 2 } n ( n + 1 ) ( 6 n - 1 )$$
for all positive integers $n$.
Given that
$$\sum _ { r = 1 } ^ { 12 } \left( 9 r ^ { 2 } - 4 r + k \left( 2 ^ { r } \right) \right) = 6630$$
(b) find the exact value of the constant $k$.\\
\hfill \mbox{\textit{Edexcel F1 2014 Q5 [8]}}