you deposit $400 each month into an account earning 6% interest compounded monthly. a) how much will you…

Explanation:

Step1: Identify the relevant formula for future - value of an ordinary annuity

The formula for the future - value of an ordinary annuity is $F = A\times\frac{(1 + r)^{n}-1}{r}$, where $A$ is the amount of each payment, $r$ is the interest rate per period, and $n$ is the number of periods. The monthly deposit $A=$400$, the annual interest rate $i = 6%=0.06$, so the monthly interest rate $r=\frac{0.06}{12}=0.005$, and the number of years $t = 15$ years. The number of periods $n=15\times12 = 180$ months.

Step2: Calculate the future - value of the annuity

Substitute the values into the formula: $F = 400\times\frac{(1 + 0.005)^{180}-1}{0.005}$ First, calculate $(1 + 0.005)^{180}$. Using the formula $a^{b}$, we have $(1 + 0.005)^{180}\approx2.4540935$. Then, $(1 + 0.005)^{180}-1\approx2.4540935 - 1=1.4540935$. $\frac{(1 + 0.005)^{180}-1}{0.005}=\frac{1.4540935}{0.005}=290.8187$. $F = 400\times290.8187=$116327.48$.

Step3: Calculate the total amount of money deposited

The total amount of money deposited is the monthly deposit amount times the number of months. The monthly deposit $A = 400$ and the number of months $n = 180$. So the total deposit $D=400\times180=$72000$.

Step4: Calculate the total interest earned

The total interest $I$ is the future - value of the account minus the total amount of money deposited. $I=F - D$. $I=116327.48-72000=$44327.48$.

Answer:

a) $116327.48 b) $72000 c) $44327.48