Question

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 ____?

Answer 1

Answer:

$ 2062.97 ( approx )

Step-by-step explanation:

Since, the amount formula for continuous compounding,

$A=Pe^{rt}$

Where, P is the principal amount,

r is the rate per period,

t is the number of years,

e is euclid number,

Here,

P = $ 2000,

r = 3.1% = 0.031,

t = 1 year,

Hence, the balance would be,

$A=2000 e^{0.031\times 1}$

$= 2000 e^{0.031}$

$=\ 2062.97100777$

$\approx \ 2062.97$

Answer 2

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

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