| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | November |
| 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 and simplify. Part (i) involves careful algebraic manipulation to find the telescoping form and numerical calculation. Part (ii) requires solving an inequality. While it demands pattern recognition and algebraic skill beyond routine exercises, the telescoping structure is a standard Further Maths technique, making it moderately challenging but not requiring deep conceptual insight. |
| Spec | 4.04f Line-plane intersection: find point8.01c Sequence behaviour: periodic, convergent, divergent, oscillating, monotonic8.01d Sequence limits: limit of nth term as n tends to infinity, steady-states |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(\left(\frac{6}{\sqrt{1}}-\frac{7}{\sqrt{3}}\right)+\left(\frac{7}{\sqrt{3}}-\frac{8}{\sqrt{7}}\right)+\ldots+\left(\frac{35}{\sqrt{871}}-\frac{36}{\sqrt{931}}\right) = 6 - \frac{36}{\sqrt{931}}\) | M1A1 | |
| \(= 4.820\) | A1 | |
| Subtotal: 3 marks | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(6 - \frac{n+6}{\sqrt{n^2+n+1}} > 4.9 \Rightarrow 0.21n^2 - 10.79n - 34.79 (> 0)\) | M1A1 | |
| \(\Rightarrow n > 54.42\ldots\) so 55 terms required | dM1A1 | |
| Subtotal: 4 marks | [4] | |
| Total: 7 marks | [7] |
## Question 4:
### Part (i):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $\left(\frac{6}{\sqrt{1}}-\frac{7}{\sqrt{3}}\right)+\left(\frac{7}{\sqrt{3}}-\frac{8}{\sqrt{7}}\right)+\ldots+\left(\frac{35}{\sqrt{871}}-\frac{36}{\sqrt{931}}\right) = 6 - \frac{36}{\sqrt{931}}$ | M1A1 | |
| $= 4.820$ | A1 | |
| **Subtotal: 3 marks** | [3] | |
### Part (ii):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $6 - \frac{n+6}{\sqrt{n^2+n+1}} > 4.9 \Rightarrow 0.21n^2 - 10.79n - 34.79 (> 0)$ | M1A1 | |
| $\Rightarrow n > 54.42\ldots$ so 55 terms required | dM1A1 | |
| **Subtotal: 4 marks** | [4] | |
| **Total: 7 marks** | [7] | |
---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 $S _ { N }$. Find\\
(i) the value of $S _ { 30 }$ correct to 3 decimal places,\\
(ii) the least value of $N$ for which $S _ { N } > 4.9$.
\hfill \mbox{\textit{CAIE FP1 2015 Q4}}