Moderate -0.3 This is a standard textbook induction proof of a well-known formula (sum of squares), followed by routine algebraic manipulation and application. Part (a) is a classic exercise requiring no insight, part (b) involves simple algebra to expand and collect terms, and part (c) is straightforward substitution. While it requires multiple steps, all techniques are routine for Further Maths students and the question provides significant scaffolding.
7. (a) Prove by induction that for \(n \in \mathbb { N }\)
$$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { n } { 6 } ( n + 1 ) ( 2 n + 1 )$$
(b) Hence show that
$$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } + 2 \right) = \frac { n } { 6 } \left( a n ^ { 2 } + b n + c \right)$$
where \(a , b\) and \(c\) are integers to be found.
(c) Using your answers to part (b), find the value of
$$\sum _ { r = 10 } ^ { 25 } \left( r ^ { 2 } + 2 \right)$$
7. (a) Prove by induction that for $n \in \mathbb { N }$
$$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { n } { 6 } ( n + 1 ) ( 2 n + 1 )$$
(b) Hence show that
$$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } + 2 \right) = \frac { n } { 6 } \left( a n ^ { 2 } + b n + c \right)$$
where $a , b$ and $c$ are integers to be found.\\
(c) Using your answers to part (b), find the value of
$$\sum _ { r = 10 } ^ { 25 } \left( r ^ { 2 } + 2 \right)$$
\hfill \mbox{\textit{Edexcel F1 2021 Q7 [11]}}