1.12: Past-paper question 8

The $\displaystyle n$th term of a geometric series $\displaystyle G$ is $\displaystyle U_n$ and the sum of the first $\displaystyle n$ terms of $\displaystyle G$ is $\displaystyle S_n$

Given that $\displaystyle U_n = \frac{25}{4} \left( \frac{3}{5} \right)^n$

(a) find the exact value of $\displaystyle U_5$

(b) Show that $\displaystyle S_n = \sum_{r=1}^n \frac{A}{B} \left( \frac{3}{5} \right)^{r-1}$ where $\displaystyle A$ and $\displaystyle B$ are integers to be found.

The sum to infinity of $\displaystyle G$ is $\displaystyle S$

(c) Find the least value of $\displaystyle n$ such that $\displaystyle S - S_n < 0.045$