Answer:
[tex]APY=\frac{r^{4}}{4}+r[/tex]
Explanation:
We were given that:
[tex]\begin{gathered} APY=(1+\frac{r}{2})^2-1 \\ r=interest\text{ rate} \end{gathered}[/tex]
We will simplify the formula as shown below:
[tex]\begin{gathered} APY=(1+\frac{r}{2})^2-1 \\ \text{Let's expand the bracket, we have:} \\ APY=(1+\frac{r}{2})^2-1 \\ APY=(1+\frac{r}{2})(1+\frac{r}{2})-1 \\ APY=1(1+\frac{r}{2})+\frac{r}{2}(1+\frac{r}{2})-1 \\ APY=1+\frac{r}{2}+\frac{r}{2}+\frac{r^4}{4}-1 \\ APY=1+r+\frac{r^4}{4}-1 \\ \text{Put like terms together, we have:} \\ APY=\frac{r^4}{4}+r+1-1 \\ APY=\frac{r^4}{4}+r \\ \\ \therefore APY=\frac{r^{4}}{4}+r \end{gathered}[/tex]
