Hi Kyla,

The interest rate is $8\%$ per year, compounded quarterly so the interest rate for one quarter is $\large \frac{8\%}{4} = \frac{0.08}{4} = \normalsize 0.02.$ You start with $\$800$ and hence at the end of the first quarter you have what you invested plus the interest it gained,

\[\$800 + 0.02 \times \$800 = \$800 \times (1 + 0.02).\]

Hence at the beginning of the second quarter you have $\$800 \times (1 + 0.02)$ to invest and hence at the end of the second quarter you have what you invested plus the interest it gained,

\[\$800 \times (1 + 0.02) + 0.02 \times \$800 \times (1 + 0.02)= \$800 \times (1 + 0.02) \times (1 + 0.02) =\$800 \times (1 + 0.02)^2 .\]

This looks too messy so let's start again using symbols rather than numbers.

Suppose you invest $P$ dollars at $100 \times r\% = r$ per year compounded $k$ times per year for $m$ years. Since the compounding is $k$ times per year the interest rate for each compounding period is $\large \frac{r}{k}.$ Hence you start with $P$ dollars and at the end of the first quarter you have what you invested plus the interest it gained,

\[P + \frac{r}{k} \times P = P \times \left(1 + \frac{r}{k} \right).\]

Hence at the beginning of the second compounding period you have $P \times \left(1 +\large \frac{r}{k} \right)$ to invest and hence at the end of the second compounding period you have what you invested plus the interest it gained,

\[P \times \left(1 + \frac{r}{k} \right) + \frac{r}{k} \times P \times \left(1 + \frac{r}{k} \right).\]

Notice that there is a common factor of $P \times \left(1 + \large \frac{r}{k} \right)$ in the two terms above so

\[P \times \left(1 + \frac{r}{k} \right) + \frac{r}{k} \times P \times \left(1 + \frac{r}{k} \right) = P \times \left(1 + \frac{r}{k} \right) \times \left(1 + \frac{r}{k} \right) = P \times \left(1 + \frac{r}{k} \right)^2 .\]

I hope now you see the pattern. In each compounding period you multiply what you already have by $\left(1 + \large \frac{r}{k} \right).$ You continue for $m$ years, compounding $k$ times each year and thus you have $k \times m$ compounding periods and at the end your original investment of $P$ dollars has grown to

\[P \times \left(1 + \frac{r}{k} \right)^{km} \mbox{ dollars.}\]

I hope this helps and good luck in your correspondents course,
Penny