Standard +0.8 This is a two-part Further Maths induction question requiring a standard proof followed by algebraic manipulation using given formulae to derive a new result. While the induction itself is routine, the second part requires careful algebraic rearrangement and factorization to identify the quadratic Q(n), making it moderately challenging but still within typical FP1 scope.
7 Prove by induction that
$$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 5 } + r ^ { 3 } \right) = \frac { 1 } { 2 } n ^ { 3 } ( n + 1 ) ^ { 3 }$$
for all \(n \geqslant 1\).
Use this result together with the List of Formulae (MF10) to prove that
$$\sum _ { r = 1 } ^ { n } r ^ { 5 } = \frac { 1 } { 12 } n ^ { 2 } ( n + 1 ) ^ { 2 } \mathrm { Q } ( n )$$
where \(\mathrm { Q } ( n )\) is a quadratic function of \(n\) which is to be determined.
7 Prove by induction that
$$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 5 } + r ^ { 3 } \right) = \frac { 1 } { 2 } n ^ { 3 } ( n + 1 ) ^ { 3 }$$
for all $n \geqslant 1$.
Use this result together with the List of Formulae (MF10) to prove that
$$\sum _ { r = 1 } ^ { n } r ^ { 5 } = \frac { 1 } { 12 } n ^ { 2 } ( n + 1 ) ^ { 2 } \mathrm { Q } ( n )$$
where $\mathrm { Q } ( n )$ is a quadratic function of $n$ which is to be determined.
\hfill \mbox{\textit{CAIE FP1 2008 Q7 [8]}}