Dylan deposits $\$ 5,800$ every year into an account earning an annual interest rate of $6.8 \%$ compounded annually. How much would he have in the account after $6$ years, 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:

Full solution

Q. Dylan deposits

\$ 5,800

every year into an account earning an annual interest rate of

6.8 \%

compounded annually. How much would he have in the account after

6

years, 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 Variables: Identify the variables from the problem to use in the formula. $\newline$ We have: $\newline$ $d$ (the amount invested at the end of each period) = $\$5,800$ $\newline$ $i$ (the interest rate per period) = $6.8\%$ or $0.068$ when converted to decimal $\newline$ $n$ (the number of periods) = $6$ years
Convert Interest Rate: Convert the interest rate from a percentage to a decimal. $i = 6.8\% = \frac{6.8}{100} = 0.068$
Plug Values into Formula: Plug the values into the formula to calculate the future value of the account. $\newline$ $A = d \times \left(\frac{(1 + i)^n - 1}{i}\right)$ $\newline$ $A = 5800 \times \left(\frac{(1 + 0.068)^6 - 1}{0.068}\right)$
Calculate Value Inside Parentheses: Calculate the value inside the parentheses. $\newline$ Calculate $(1 + i)^n$ : $\newline$ $(1 + 0.068)^6 = 1.068^6$
Calculate Exponent: Calculate the exponent part of the formula. $\newline$ $1.068^6 \approx 1.4843$ (rounded to four decimal places for intermediate calculations)
Subtract One: Subtract $1$ from the result of Step $5$ . $\newline$ $1.4843 - 1 = 0.4843$
Divide by i: Divide the result of Step $6$ by $i$ . $\newline$ $0.4843 / 0.068 \approx 7.1250$ (rounded to four decimal places for intermediate calculations)
Multiply by $d$ : Multiply the result of Step $7$ by $d$ to find the future value $A$ . $A = 5800 \times 7.1250$ $A \approx 41325$ (rounded to the nearest dollar)