| Exam Board | OCR |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and series, recurrence and convergence |
| Type | Recurrence relation solving for closed form |
| Difficulty | Standard +0.3 This is a standard FP1 recurrence relation question with routine calculation followed by pattern-spotting and proof by induction. The recurrence is straightforward to iterate, the pattern (u_n = 2/(1+2n)) emerges clearly from the given values, and the induction proof is mechanical with simple algebraic manipulation. While it requires multiple techniques, each step follows a well-practiced template with no novel insight needed. |
| Spec | 4.01a Mathematical induction: construct proofs |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(\frac{2}{3}, \frac{2}{5}, \frac{2}{7}\) | B1, B1, B1 [3] | B1 x 3, Obtain 3 correct values; Justify given answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(\frac{2}{2n-1}\) | M1, A1 [2] | Fraction, in terms of n, with correct numerator or denominator; Obtain correct answer a.e.f. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(\frac{2}{2(n+1)-1}\) | B1ft, M1, A1, A1, B1 [5] | Verify result true when \(n=1\), for their (ii), or \(n=2, 3\) or \(4\); Expression for \(u_{n+1}\) using recurrence relation in terms of n using their (ii); Correct unsimplified answer; Correct answer in correct form; Specific statement of induction conclusion, previous 4 marks must be earned, n=1 must be verified |
### (i)
Answer: $\frac{2}{3}, \frac{2}{5}, \frac{2}{7}$ | B1, B1, B1 [3] | B1 x 3, Obtain 3 correct values; Justify given answer
### (ii)
Answer: $\frac{2}{2n-1}$ | M1, A1 [2] | Fraction, in terms of n, with correct numerator or denominator; Obtain correct answer a.e.f.
### (iii)
Answer: $\frac{2}{2(n+1)-1}$ | B1ft, M1, A1, A1, B1 [5] | Verify result true when $n=1$, for their (ii), or $n=2, 3$ or $4$; Expression for $u_{n+1}$ using recurrence relation in terms of n using their (ii); Correct unsimplified answer; Correct answer in correct form; Specific statement of induction conclusion, previous 4 marks must be earned, n=1 must be verifiedThe sequence $u_1, u_2, u_3, \ldots$ is defined by $u_1 = 2$ and $u_{n+1} = \frac{u_n}{1 + u_n}$ for $n \geq 1$.
\begin{enumerate}[label=(\roman*)]
\item Find $u_2$ and $u_3$, and show that $u_4 = \frac{2}{7}$. [3]
\item Hence suggest an expression for $u_n$. [2]
\item Use induction to prove that your answer to part (ii) is correct. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR FP1 2013 Q10 [10]}}