| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Sum from n+1 to 2n or similar range |
| Difficulty | Standard +0.3 This is a straightforward method of differences question with clear guidance. Part (a) is routine algebraic verification requiring expansion of (r+1)³ terms. Part (b) applies the standard telescoping technique with given endpoints. While it requires careful bookkeeping with the sum limits (50 to 99), the method is explicitly stated and the algebraic manipulation is mechanical for Further Maths students. |
| Spec | 4.06b Method of differences: telescoping series |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(f(r+1) - f(r) = r(r+1)^2 - (r-1)r^2\) | M1 | |
| \(= r(r^2 + 2r + 1 - r^2 + r)\) | A1 | any expanded form |
| \(= r(3r+1)\) | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(r=50\): \(f(51)-f(50)\); \(r=51\): \(f(52)-f(51)\); \(r=99\): \(f(100)-f(99)\) | M1A1 | PI, clearly shown. Accept \(\sum_1^{99} - \sum_1^{49}\) |
| \(\sum_{r=50}^{99} r(3r+1) = f(100) - f(50)\) | m1 | clear cancellation |
| \(= 867500\) | A1F | cao |
## Question 1:
### Part (a)
| Working | Marks | Guidance |
|---------|-------|----------|
| $f(r+1) - f(r) = r(r+1)^2 - (r-1)r^2$ | M1 | |
| $= r(r^2 + 2r + 1 - r^2 + r)$ | A1 | any expanded form |
| $= r(3r+1)$ | A1 | AG |
### Part (b)
| Working | Marks | Guidance |
|---------|-------|----------|
| $r=50$: $f(51)-f(50)$; $r=51$: $f(52)-f(51)$; $r=99$: $f(100)-f(99)$ | M1A1 | PI, clearly shown. Accept $\sum_1^{99} - \sum_1^{49}$ |
| $\sum_{r=50}^{99} r(3r+1) = f(100) - f(50)$ | m1 | clear cancellation |
| $= 867500$ | A1F | cao |
---1
\begin{enumerate}[label=(\alph*)]
\item Given that $\mathrm { f } ( r ) = ( r - 1 ) r ^ { 2 }$, show that
$$\mathrm { f } ( r + 1 ) - \mathrm { f } ( r ) = r ( 3 r + 1 )$$
\item Use the method of differences to find the value of
$$\sum _ { r = 50 } ^ { 99 } r ( 3 r + 1 )$$
(4 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2007 Q1 [7]}}