9 For the sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), it is given that \(u _ { 1 } = 8\) and
$$u _ { r + 1 } = \frac { 5 u _ { r } - 3 } { 4 }$$
for all \(r\).
- Prove by mathematical induction that
$$u _ { n } = 4 \left( \frac { 5 } { 4 } \right) ^ { n } + 3$$
for all positive integers \(n\).
- Deduce the set of values of \(x\) for which the infinite series
$$\left( u _ { 1 } - 3 \right) x + \left( u _ { 2 } - 3 \right) x ^ { 2 } + \ldots + \left( u _ { r } - 3 \right) x ^ { r } + \ldots$$
is convergent.
- Use the result given in part (i) to find surds \(a\) and \(b\) such that
$$\sum _ { n = 1 } ^ { N } \ln \left( u _ { n } - 3 \right) = N ^ { 2 } \ln a + N \ln b .$$