Question

Lesotho is a country in southern Africa whose geographical boundaries lie entirely within those of South Africa. In 2010, Lesotho's real GDP per capita was $300. Suppose that Lesotho's real GDP per capita averages a growth rate of 4 percent per year. In what year will Lesotho's real GDP per capita be approximately $1,200?

Answer 1

you have
\[ S_0=300\]
then
\[ S_1=300+300*4\%=300(1+4\%)=300*(1+0.04) \]
then
\[ S_2=S_1+4\%*S_1=S_1(1+0.04)=300(1+0.04)^2 \]
so in general
\[ S_n=300\cdot(1+0.04)^ n \]

Answer 2

Now, you have to solve the equation
\[ 1200=300\cdot(1+0.04)^n \]

Answer 3

first, divide by 300 so we have
\[ 4=(1+0.04)^n=1.04^ n \]
applying logarithms:
\[ \ln4=\ln1.04^n=n\cdot\ln1.04 \]
from this
\[ n=\frac{\ln 4}{\ln 1.04}\approx 35.35 \]

So, in 35+ years Lesotho's real GDP per capita will be $1200.