A painting, purchased for $10 000 in 1990, increased in value by 8% per year. How do you find the value of the painting in the year 2000?

Let's try to find the general law of this.

The initial value of the paint, at zero years, is #V_0# (later we will substitute the $10,000).

After the first year we have a new value that is augmented of the percentage given. We call the percentage P (later we will substitute 8/100).

#V_1=V_0+P*V_0=V_0(P+1)#.

The second year we have the same rule, but applied to #V_1#

#V_2=V_1+P*V_1=V_1(P+1)#

and substituting the previous value of #V_1=V_0(P+1)# we have

#V_2=V_0(P+1)(P+1)=V_0(P+1)^2#.

It seems we found a rule, let's try again for the third year

#V_3=V_2+P*V_2=V_2(P+1)#

and substituting the previous equation for #V_2=V_0(P+1)^2#

#V_3=V_0(P+1)^2(P+1)=V_0(P+1)^3#.

Then, after #N# years, we have

#V_N=V_0(P+1)^N#.

This is the rule that is governing the increases of the value in the years. Now it is enough to substitute

#V_0=10000#

#P=8/100#

#N=2000-1990=10#

#V_N=10000(8/100+1)^10#

#=10000*(108/100)^10#

#=10000*(1.08)^10#

#\approx10000*2.16#

#=21600#.

This paint was a good investment, after #10# years it has a value that is more than double: $21 600.