Standard +0.3 This is a straightforward proof by induction on a linear recurrence relation. The formula is given, requiring only verification of the base case and standard algebraic manipulation in the inductive step. While it's a Further Maths topic, the mechanics are routine with no conceptual surprises, making it slightly easier than average overall but typical for FP1 induction questions.
6 A sequence is defined by \(u _ { 1 } = 3\) and \(u _ { n + 1 } = 3 u _ { n } - 5\). Prove by induction that \(u _ { n } = \frac { 3 ^ { n - 1 } + 5 } { 2 }\).
Section B (36 marks)
\(u_1 = 3\) and \(\frac{3^{1-1}+5}{2} = 3\), so true for \(n=1\)
B1
Must show working on given result with \(n=1\)
Assume true for \(n=k\): \(\Rightarrow u_k = \dfrac{3^{k-1}+5}{2}\)
E1
Assuming true for k. Allow "Let \(n=k\) and (result)" or "If \(n=k\) and (result)". Do not allow "\(n=k\)" or "Let \(n=k\)" without the result quoted, followed by working
\(u_{k+1}\) with substitution of result for \(u_k\) and some working to follow
\(= \dfrac{3^k + 15}{2} - 5\)
\(= \dfrac{3^k + 15 - 10}{2}\)
\(= \dfrac{3^k + 5}{2}\)
A1
Correctly obtained
\(= \dfrac{3^{n-1}+5}{2}\) when \(n = k+1\)
Or target seen
Therefore if true for \(n=k\) it is also true for \(n=k+1\)
E1
Both points explicit. Dependent on A1 and previous E1
Since it is true for \(n=1\), it is true for all positive integers n
E1
Dependent on B1 and previous E1
[6]
## Question 6:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $u_1 = 3$ and $\frac{3^{1-1}+5}{2} = 3$, so true for $n=1$ | B1 | Must show working on given result with $n=1$ |
| Assume true for $n=k$: $\Rightarrow u_k = \dfrac{3^{k-1}+5}{2}$ | E1 | Assuming true for k. Allow "Let $n=k$ and (result)" or "If $n=k$ and (result)". Do not allow "$n=k$" or "Let $n=k$" without the result quoted, followed by working |
| $\Rightarrow u_{k+1} = 3\left(\dfrac{3^{k-1}+5}{2}\right) - 5$ | M1 | $u_{k+1}$ with substitution of result for $u_k$ and some working to follow |
| $= \dfrac{3^k + 15}{2} - 5$ | | |
| $= \dfrac{3^k + 15 - 10}{2}$ | | |
| $= \dfrac{3^k + 5}{2}$ | A1 | Correctly obtained |
| $= \dfrac{3^{n-1}+5}{2}$ when $n = k+1$ | | Or target seen |
| Therefore if true for $n=k$ it is also true for $n=k+1$ | E1 | Both points explicit. Dependent on A1 and previous E1 |
| Since it is true for $n=1$, it is true for all positive integers n | E1 | Dependent on B1 and previous E1 |
| | **[6]** | |
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6 A sequence is defined by $u _ { 1 } = 3$ and $u _ { n + 1 } = 3 u _ { n } - 5$. Prove by induction that $u _ { n } = \frac { 3 ^ { n - 1 } + 5 } { 2 }$.
Section B (36 marks)
\hfill \mbox{\textit{OCR MEI FP1 2015 Q6 [6]}}