Question

4. An initial investment of S4 is worth $100 after 5 years. If the annual growth reflects a geometric sequence, approximately how much will the investment be worth after 11 years?•220•5590•244•12500

Answer 1

The formula for geometric sequence is,

$\begin{gathered} U_n=ar^{n-1} \\ \end{gathered}$

Where,

$\begin{gathered} U_5=\text{ \ 100} \\ n=5\text{years} \\ a=\text{ \ 4} \end{gathered}$

Substituting the values into the formula above,

$\begin{gathered} U_5=ar^{5-1}=ar^4 \\ 4r^4=100 \\ r^4=\frac{100}{4} \\ r^4=25 \\ \sqrt[4]{r^4}=\sqrt[4]{25} \\ r=\sqrt[4]{25} \end{gathered}$

After 11 years,

$\begin{gathered} U_{11}=ar^{11-1}=ar^{10} \\ U_{11}=ar^{10} \\ U_{11}=4(\sqrt[4]{25})^{10} \\ U_{11}=4\times3125=12500 \end{gathered}$

Hence, the amount after 11 years is $12,500