| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Finding n for given sum value |
| Difficulty | Challenging +1.2 This is a telescoping series question requiring students to recognize the pattern, simplify consecutive terms to show cancellation, and then perform straightforward calculations. While it's a Further Maths topic and requires some algebraic manipulation to identify the telescoping nature, the actual execution is mechanical once the pattern is spotted. The second part involves simple iteration or inequality solving. |
| Spec | 4.06b Method of differences: telescoping series |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left(\frac{6}{\sqrt{1}}-\frac{7}{\sqrt{3}}\right)+\left(\frac{7}{\sqrt{3}}-\frac{8}{\sqrt{7}}\right)+\ldots+\left(\frac{35}{\sqrt{871}}-\frac{36}{\sqrt{931}}\right) = 6 - \frac{36}{\sqrt{931}} = 4.820\) | 3 | M1A1A1 |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(6 - \frac{n+6}{\sqrt{n^2+n+1}} > 4.9 \Rightarrow 0.21n^2 - 10.79n - 34.79 (> 0)\) | 2 | M1*A1 |
| \(\Rightarrow n > 54.42\ldots\) so 55 terms required | 2 | DM1A1 |
| Total | 4 |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left(\frac{6}{\sqrt{1}}-\frac{7}{\sqrt{3}}\right)+\left(\frac{7}{\sqrt{3}}-\frac{8}{\sqrt{7}}\right)+\ldots+\left(\frac{35}{\sqrt{871}}-\frac{36}{\sqrt{931}}\right) = 6 - \frac{36}{\sqrt{931}} = 4.820$ | 3 | M1A1A1 |
| **Total** | **3** | |
## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $6 - \frac{n+6}{\sqrt{n^2+n+1}} > 4.9 \Rightarrow 0.21n^2 - 10.79n - 34.79 (> 0)$ | 2 | M1*A1 |
| $\Rightarrow n > 54.42\ldots$ so 55 terms required | 2 | DM1A1 |
| **Total** | **4** | |
---4 The sequence $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is such that, for all positive integers $n$,
$$a _ { n } = \frac { n + 5 } { \sqrt { } \left( n ^ { 2 } - n + 1 \right) } - \frac { n + 6 } { \sqrt { } \left( n ^ { 2 } + n + 1 \right) }$$
The sum $\sum _ { n = 1 } ^ { N } a _ { n }$ is denoted by $S _ { N }$.\\
(i) Find the value of $S _ { 30 }$ correct to 3 decimal places.\\
(ii) Find the least value of $N$ for which $S _ { N } > 4.9$.\\
\hfill \mbox{\textit{CAIE FP1 2017 Q4 [7]}}