The sum to infinity of a geometric series is 96 The first term of the series is less than 30 The second term of the series is 18 8 (a) Find the first term and common ratio of the series - AQA - A-Level Maths Mechanics - Question 8 - 2020 - Paper 3

Starting from:

$u_n = \frac{3^{n}}{2^{n - 5}}$

We apply the logarithmic identity:

$\log_{b} \left( \frac{m}{n} \right) = \log_{b} m - \log_{b} n$

Thus, we have:

$\log_{3} u_n = \log_{3} (3^{n}) - \log_{3} (2^{n - 5})$

Calculating these logarithms:

$= n - (n - 5) \log_{3} 2$

This simplifies to:

$= n - n \log_{3} 2 + 5 \log_{3} 2$

Now we combine the terms:

$= n(1 - 2 \log_{3} 2) + 5 \log_{3} 2$

Thus, we confirm:

$\log_{3} u_n = n(1 - 2 \log_{2} 2) + 5 \log_{3} 2$.