| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Finding constants from given sum formula |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question requiring algebraic manipulation of standard summation formulas. Students expand r²(r+2), apply known formulas for Σr² and Σr³, then factor to match the given form and identify constants by comparing coefficients—routine technique with no novel insight required. |
| Spec | 4.06a Summation formulae: sum of r, r^2, r^3 |
\begin{enumerate}
\item Use the standard results for $\sum _ { r = 1 } ^ { n } r ^ { 2 }$ and $\sum _ { r = 1 } ^ { n } r ^ { 3 }$ to show that, for all positive integers $n$
\end{enumerate}
$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 2 ) = \frac { 1 } { 12 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$
where $a$, $b$ and $c$ are integers to be determined.
\hfill \mbox{\textit{Edexcel F1 2023 Q1 [4]}}