Use the formula for continuous compounding to compute the balance in the account after​ 1, 5, and 20 years.​ Also, find the APY for the account. A ​$2000 deposit in an account with an APR of 3.1​%. The balance in the account after 1 year is approximately ____?

[tex]\bf \qquad \textit{Continuously Compounding Interest Earned Amount}\\\\ A=Pe^{rt}\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\to& \$2000\\ r=rate\to 3.1\%\to \frac{3.1}{100}\to &0.031\\ t=years\to &1,5,20 \end{cases} \\\\\\ \stackrel{\textit{for 1 year}}{A=2000e^{0.031\cdot 1}}\qquad \stackrel{\textit{for 5 years}}{A=2000e^{0.031\cdot 5}}\qquad \stackrel{\textit{for 20 years}}{A=2000e^{0.031\cdot 20}}\\\\ -------------------------------\\\\[/tex]

[tex]\bf \qquad \qquad \textit{Annual Yield Formula} \\\\ ~~~~~~~~~\left(1+\frac{r}{n}\right)^{n}-1 \\\\ \begin{cases} r=rate\to 3.1\%\to \frac{3.1}{100}\to &0.031\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{continuously, 365 days then} \end{array}\to &365 \end{cases} \\\\\\ \left(1+\frac{0.031}{365}\right)^{365}-1 \approx 0.03148414 \\\\\\ 0.03148414\cdot 100\implies 3.148\% \approx 3.15\%[/tex]