Joseph is saving money and plans on making monthly contributions into an account earning a monthly interest rate of $0.75 \%$ . If Joseph would like to end up with $\$ 259,000$ after $13$ years, how much does he need to contribute to the account every month, to the nearest dollar? Use the following formula to determine your answer. $\newline$ $A=d\left(\frac{(1+i)^{n}-1}{i}\right)$ $\newline$ $A=$ the future value of the account after $n$ periods $\newline$ $d=$ the amount invested at the end of each period $\newline$ $i=$ the interest rate per period $\newline$ $n=$ the number of periods $\newline$ Answer:

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Compound interest

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Q. Joseph is saving money and plans on making monthly contributions into an account earning a monthly interest rate of

0.75 \%

. If Joseph would like to end up with

\$ 259,000

after

13

years, how much does he need to contribute to the account every month, to the nearest dollar? Use the following formula to determine your answer.

\newline

A=d\left(\frac{(1+i)^{n}-1}{i}\right)

\newline

A=

the future value of the account after

n

periods

\newline

d=

the amount invested at the end of each period

\newline

i=

the interest rate per period

\newline

n=

the number of periods

\newline

Answer:

Identify Given Values: Identify the given values from the problem. $\newline$ $A$ (future value of the account) = $\$259,000$ $\newline$ $i$ (monthly interest rate) = $0.75\%$ or $0.0075$ when converted to decimal $\newline$ $n$ (total number of periods in months) = $13$ years $* 12$ months/year = $156$ months $\newline$ We will use the formula $A=d\left(\frac{(1+i)^{n}-1}{i}\right)$ to find $\$259,000$ $0$ , the monthly contribution.
Substitute Values into Formula: Substitute the given values into the formula. $\newline$ $A = \$259,000$ $\newline$ $i = 0.0075$ $\newline$ $n = 156$ $\newline$ Now, plug these values into the formula to solve for $d$ .
Calculate Exponent: Calculate the value inside the parentheses and the exponent. $\newline$ First, calculate $(1+i)^n$ which is $(1+0.0075)^{156}$ .
Compute $(1+0.0075)^{156}$ : Use a calculator to compute $(1+0.0075)^{156}$ . $(1+0.0075)^{156} \approx 3.4449$ (rounded to four decimal places for simplicity)
Subtract $1$ : Subtract $1$ from the result obtained in Step $4$ . $\newline$ $3.4449 - 1 \approx 2.4449$
Divide by $i$ : Divide the result from Step $5$ by $i$ . $\frac{2.4449}{0.0075} \approx 325.9867$
Substitute into Formula: Substitute the result from Step $6$ into the formula to solve for $d$ .
$\$259,000 = d \times 325.9867$
Now, solve for $d$ by dividing both sides of the equation by $325.9867$ .
Calculate Monthly Contribution: Calculate the monthly contribution $d$ . $\newline$ $d = \frac{\$259,000}{325.9867} \approx \$794.55$ $\newline$ Since the question asks for the nearest dollar, we round this to $\$795$ .