Question

In order for an investment, which is increasing in value exponentially, to increase by a factor of 5 in 20

years, about what percent does it need to grow each year?

Answer 1

we can look at it like this hmmm

what APR is needed for one dolla! to turn into 5 dolla! in 20 years?

`\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 5\\ P=\textit{initial amount}\dotfill &\$1\\ r=rate\to r\%\to (r)/(100)\\ t=years\dotfill &20\\ \end{cases} \\\\\\ 5 = 1(1 + (r)/(100))^(20) \implies 5=\left( \cfrac{100+r}{100} \right)^(20)\implies \sqrt[20]{5}=\cfrac{100+r}{100} \\\\\\ 100\sqrt[20]{5}=100+r\implies 100\sqrt[20]{5}-100=r\implies \stackrel{ \% }{8.34}\approx r`