#### Answer

a. The value at the end of the fifth year is
$a_{5} \approx 3932$
b. The value at the end of the eighth year is
$a_{5} \approx 2013$

#### Work Step by Step

Since the machine loses $20\%$ of its value each year, it retains $80\%$ of its value
Its value at the end of each of the following years is $80\%$ of the previous year’s value. These values form a geometric sequence, with $r = 0.8, a=12000$
a. The value at the end of the fifth year is
$a_{5}=12.000(0.8)^{6-1} \approx 3932$
b. The value at the end of the eighth year is
$a_{5}=12.000(0.8)^{9-1} \approx 2013$