We know that the painting increase its value by 5% each year.
So, if P(1) is the value the next year and P(0) is the actual value ($27,400) we can write:
[tex]P(1)_{}=P(0)+0.05P(0)=1.05\cdot P(0)[/tex]In the same way, the following year, it will increase another 5% over its value:
[tex]P(2)=1.05P(1)=1.05(1.05\cdot P(0))=1.05^2\cdot P(0)=1.05^2\cdot27,400[/tex]We can generalize this as:
[tex]P(n)=27,400\cdot1.05^n[/tex]For n=3 (3 years) we will have a value of:
[tex]P(3)=27,400\cdot1.05^3\approx27,400\cdot1.1576\approx31,718.93[/tex]Answer: the value of the painting in 3 years is expected to be $31,718.93.
