OCR MEI FP1 2011 January — Question 6 6 marks

Exam BoardOCR MEI
ModuleFP1 (Further Pure Mathematics 1)
Year2011
SessionJanuary
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProof by induction
TypeProve recurrence relation formula
DifficultyStandard +0.3 This is a straightforward proof by induction with a simple recurrence relation. The base case is trivial (u₁ = 5 = 2² + 1), and the inductive step requires only basic algebraic manipulation of powers of 2. While it's a Further Maths topic, it's a standard textbook exercise requiring no novel insight—slightly easier than average overall but typical for FP1 induction questions.
Spec4.01a Mathematical induction: construct proofs
6 A sequence is defined by \(u _ { 1 } = 5\) and \(u _ { n + 1 } = u _ { n } + 2 ^ { n + 1 }\). Prove by induction that \(u _ { n } = 2 ^ { n + 1 } + 1\).
Question 6:
AnswerMarks Guidance
Answer/WorkingMark Guidance
When \(n=1\), \(2^{1+1}+1=5\), so true for \(n=1\)B1
Assume \(u_k = 2^{k+1}+1\)E1 Assuming true for \(k\)
\(\Rightarrow u_{k+1} = 2^{k+1}+1+2^{k+1}\)M1 Using \(u_k\) to find \(u_{k+1}\)
\(= 2 \times 2^{k+1}+1 = 2^{k+2}+1 = 2^{(k+1)+1}+1\)A1 Correct simplification
Therefore if true for \(k\) it is also true for \(k+1\)E1 Dependent on A1 and previous E1
Since true for \(n=1\), true for all positive integersE1 [6] Dependent on B1 and previous E1 
# Question 6:

| Answer/Working | Mark | Guidance |
|---|---|---|
| When $n=1$, $2^{1+1}+1=5$, so true for $n=1$ | B1 | |
| Assume $u_k = 2^{k+1}+1$ | E1 | Assuming true for $k$ |
| $\Rightarrow u_{k+1} = 2^{k+1}+1+2^{k+1}$ | M1 | Using $u_k$ to find $u_{k+1}$ |
| $= 2 \times 2^{k+1}+1 = 2^{k+2}+1 = 2^{(k+1)+1}+1$ | A1 | Correct simplification |
| Therefore if true for $k$ it is also true for $k+1$ | E1 | Dependent on A1 and previous E1 |
| Since true for $n=1$, true for all positive integers | E1 [6] | Dependent on B1 and previous E1 |

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6 A sequence is defined by $u _ { 1 } = 5$ and $u _ { n + 1 } = u _ { n } + 2 ^ { n + 1 }$. Prove by induction that $u _ { n } = 2 ^ { n + 1 } + 1$.

\hfill \mbox{\textit{OCR MEI FP1 2011 Q6 [6]}}
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