you deposit $100 each year into an account ea...
you deposit $100 each year into an account earning 6% compounded annually. do not round numbers while working on the problem. only round the final answer. since we are working with money, round to the cent. a) how much will you have in the account in 15 years? b) how much total money will you put into the account? c) how much total interest will you earn?
Answer
# Explanation:
## Step1: Identify the 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 annual payment, $r$ is the interest rate per period, and $n$ is the number of periods. Here, $A = 100$, $r=0.06$, and $n = 15$.
## Step2: Calculate the future - value of the annuity
Substitute the values into the formula:
\[
\begin{align*}
F&=100\times\frac{(1 + 0.06)^{15}-1}{0.06}\\
&=100\times\frac{1.06^{15}-1}{0.06}\\
\end{align*}
\]
First, calculate $1.06^{15}\approx2.396558$. Then $1.06^{15}-1\approx1.396558$. And $\frac{1.06^{15}-1}{0.06}\approx\frac{1.396558}{0.06}\approx23.27597$. So $F = 100\times23.27597=2327.597\approx2327.60$.
## Step3: Calculate the total amount of money deposited
The annual deposit is $A = 100$, and the number of years $n = 15$. The total amount of money deposited $T_d=A\times n=100\times15 = 1500$.
## Step4: Calculate the total interest earned
The total interest earned $T_i=F - T_d$. Substitute $F = 2327.60$ and $T_d = 1500$ into the formula. So $T_i=2327.60 - 1500=827.60$.
# Answer:
a) $2327.60
b) $1500
c) $827.60