Question 5 - A-Level Maths

Exam Board	AQA
Module	C2 (Core Mathematics 2)
Year	2005
Session	June
Topic	Geometric Sequences and Series

5 The sum to infinity of a geometric series is four times the first term of the series.

Show that the common ratio, $r$, of the geometric series is $\frac { 3 } { 4 }$.
The first term of the geometric series is 48 . Find the sum of the first 10 terms of the series, giving your answer to four decimal places.
The $n$th term of the geometric series is $u _ { n }$ and the ( $2 n$ )th term of the series is $u _ { 2 n }$.
1. Write $u _ { n }$ and $u _ { 2 n }$ in terms of $n$.
2. Hence show that $\log _ { 10 } \left( u _ { n } \right) - \log _ { 10 } \left( u _ { 2 n } \right) = n \log _ { 10 } \left( \frac { 4 } { 3 } \right)$.
3. Hence show that the value of $$\log _ { 10 } \left( \frac { u _ { 100 } } { u _ { 200 } } \right)$$ is 12.5 correct to three significant figures.