Depreciation as a geometric progression

Statement

A car’s value at the end of the year is $80\%$ of its value at the start of the year. If it costs $18\,000\ \text{€}$, what is it worth at the end of the fifth year?

Solution

The end-of-year values form a geometric progression with ratio $q = 0{.}8$: $a_n = 18\,000\cdot 0{.}8^{\,n}$. $a_5 = 18\,000\cdot 0{.}8^{5} = 18\,000\cdot 0{.}32768 = \boxed{5898{.}24\ \text{€}}.$