8 In this question you must show detailed reasoning.
A sequence of vectors \(\mathbf { a } _ { 1 } , \mathbf { a } _ { 2 } , \mathbf { a } _ { 3 } , \ldots\) is defined by
- \(\mathbf { a } _ { 1 } = \left( \begin{array} { l } 1
1
1 \end{array} \right)\) - \(\quad \mathbf { a } _ { n + 1 } = \left( \mathbf { a } _ { n } \times \mathbf { b } \right) \times \mathbf { b }\), for integers \(n \geqslant 1\), where \(\mathbf { b }\) is the vector \(\frac { 1 } { 4 } \left( \begin{array} { c } - 3
1
2 \end{array} \right)\).
- Prove by induction that \(\mathbf { a } _ { n } = \left( - \frac { 7 } { 8 } \right) ^ { n - 1 } \left( \begin{array} { l } 1
1
1 \end{array} \right)\). for all integers \(n \geqslant 1\). - Use an algebraic method to find the smallest value of \(n\) such that \(\left| \mathbf { a } _ { n } \right| < 0.001\).
\section*{END OF QUESTION PAPER}