OCR FP1 2008 June — Question 4 6 marks

Exam BoardOCR
ModuleFP1 (Further Pure Mathematics 1)
Year2008
SessionJune
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProof by induction
TypeProve matrix power formula
DifficultyStandard +0.8 This is a standard Further Maths proof by induction involving matrix powers. While it requires careful matrix multiplication and algebraic manipulation of the (1,2) entry involving powers of 3, the structure is routine: verify base case, assume for n=k, prove for n=k+1. The algebra is moderately involved but follows a predictable template for FP1 students.
Spec4.01a Mathematical induction: construct proofs4.03b Matrix operations: addition, multiplication, scalar

4 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 3 & 1 \\ 0 & 1 \end{array} \right)\). Prove by induction that, for \(n \geqslant 1\), $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 3 ^ { n } & \frac { 1 } { 2 } \left( 3 ^ { n } - 1 \right) \\ 0 & 1 \end{array} \right)$$

4 The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l } 3 & 1 \\ 0 & 1 \end{array} \right)$. Prove by induction that, for $n \geqslant 1$,

$$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 
3 ^ { n } & \frac { 1 } { 2 } \left( 3 ^ { n } - 1 \right) \\
0 & 1
\end{array} \right)$$

\hfill \mbox{\textit{OCR FP1 2008 Q4 [6]}}