| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Finding constants from given sum formula |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring algebraic manipulation of series formulae, solving for constants through expansion and comparison, then applying the result to find an unknown limit. While the techniques are standard for FM students (expanding summations, using geometric series), the multi-step nature and need to work backwards from a given form elevates it above average difficulty. |
| Spec | 4.06a Summation formulae: sum of r, r^2, r^34.06b Method of differences: telescoping series |
\begin{enumerate}
\item (a) 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}
$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 1 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( a n + b )$$
where $a$ and $b$ are integers to be determined.\\
(b) Given that
$$\sum _ { r = 5 } ^ { 25 } r ^ { 2 } ( r + 1 ) + \sum _ { r = 1 } ^ { k } 3 ^ { r } = 140543$$
find the value of the integer $k$.
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\hfill \mbox{\textit{Edexcel F1 2018 Q3 [8]}}