| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure AS (Further Additional Pure AS) |
| Year | 2019 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and Series |
| Type | Convergence and Limits of Sequences |
| Difficulty | Challenging +1.2 This is a Further Maths question on recursive sequences and limits, requiring students to set L = √L + 4/√L and solve the resulting equation. While it involves Further Maths content (convergence of sequences), the technique is standard: assume convergence to limit L, substitute into the recurrence relation, and solve algebraically. Part (b) is a straightforward extension using the same method with a parameter. The algebraic manipulation is routine for Further Maths students, making this moderately above average difficulty but not requiring novel insight. |
| Spec | 8.01d Sequence limits: limit of nth term as n tends to infinity, steady-states |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (a) | Limit = 4 |
| [1] | 1.1 | BC |
| (b) | Setting b = b = 9 throughout |
| Answer | Marks |
|---|---|
| k = 18 | M1 |
| Answer | Marks |
|---|---|
| A1 | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | soi |
| Answer | Marks |
|---|---|
| n | M1 |
| Evidence of systematic approach (e.g. k = 9 6.11…, k = 20 9.56…) | M1 |
| k = 18 | A1 |
Question 2:
2 | (a) | Limit = 4 | B1
[1] | 1.1 | BC
(b) | Setting b = b = 9 throughout
n + 1 n
k
Solving 9 9
9
k = 18 | M1
M1
A1 | 3.1a
1.1a
1.1 | soi
BC or by inspection
Alternative method
k
Search method for b = b with various k’s
n + 1 n
b
n | M1
Evidence of systematic approach (e.g. k = 9 6.11…, k = 20 9.56…) | M1
k = 18 | A1
[3]2
\begin{enumerate}[label=(\alph*)]
\item The convergent sequence $\left\{ \mathrm { a } _ { \mathrm { n } } \right\}$ is defined by $a _ { 0 } = 1$ and $\mathrm { a } _ { \mathrm { n } + 1 } = \sqrt { \mathrm { a } _ { \mathrm { n } } } + \frac { 4 } { \sqrt { \mathrm { a } _ { \mathrm { n } } } }$ for $n \geqslant 0$. Calculate the limit of the sequence.
\item The convergent sequence $\left\{ \mathrm { b } _ { \mathrm { n } } \right\}$ is defined by $\mathrm { b } _ { 0 } = 1$ and $\mathrm { b } _ { \mathrm { n } + 1 } = \sqrt { \mathrm { b } _ { \mathrm { n } } } + \frac { \mathrm { k } } { \sqrt { \mathrm { b } _ { \mathrm { n } } } }$ for $n \geqslant 0$, where $k$ is a constant.
Determine the value of $k$ for which the limit of the sequence is 9 .
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure AS 2019 Q2 [4]}}