| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Finding constants from given sum formula |
| Difficulty | Standard +0.8 This is a Further Maths question requiring algebraic manipulation of summation formulas. Students must expand the left side, apply standard summation formulas, equate coefficients with the given cubic expression, and solve simultaneous equations. Part (b) requires using the sum formula twice (subtracting cumulative sums). While systematic, it demands careful algebraic work and understanding of summation properties beyond standard A-level. |
| Spec | 4.06a Summation formulae: sum of r, r^2, r^34.06b Method of differences: telescoping series |
7. Given that, for all positive integers $n$,
$$\sum _ { r = 1 } ^ { n } ( r + a ) ( r + b ) = \frac { 1 } { 6 } n ( 2 n + 11 ) ( n - 1 )$$
where $a$ and $b$ are constants and $a > b$,
\begin{enumerate}[label=(\alph*)]
\item find the value of $a$ and the value of $b$.
\item Find the value of
$$\sum _ { r = 9 } ^ { 20 } ( r + a ) ( r + b )$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2015 Q7 [11]}}