14. A geometric series has a first term \(a\) and a common ratio \(r\).
- Prove that the sum of the first \(n\) terms of this series is given by
$$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$
A liquid is to be stored in a barrel.
Due to evaporation, the volume of the liquid in a barrel at the end of a year is \(7 \%\) less than the volume at the start of the year.
At the start of the first year, a barrel is filled with 180 litres of the liquid.
- Show that the amount of the liquid in this barrel at the end of 5 years is approximately 125.2 litres.
At the start of each year a new identical barrel is filled with 180 litres of the liquid so that, at the end of 20 years, there are 20 barrels containing varying amounts of the liquid.
- Calculate the total amount of the liquid, to the nearest litre, in the 20 barrels at the end of 20 years.