Question

3.
Every 6 months, Reuben Lopez puts $420 into an account paying 10% compounded semiannually.
Find the account balance after 15 years.


$29,299.53

$27,904.32

$31,500.00

$29,315.00

Answer 1

To solve this we are going to use the future value of annuity due formula: $FV=(1+ \frac{r}{n} )*P[ \frac{(1+ \frac{r}{n})^{kt}-1 }{ \frac{r}{n} } ]$
where
$FV$ is the future value
$P$ is the periodic deposit 
$r$ is the interest rate in decimal form 
$n$ is the number of times the interest is compounded per year
$k$ is the number of deposits per year

We know for our problem that $P=420$ and $t=15$. To convert the interest rate to decimal form, we are going to divide the rate by 100%: $r= \frac{10}{100} =0.1$. Since Ruben makes the deposits every 6 months, $k=2$. The interest is compounded semiannually, so 2 times per year; therefore, $k=2$.
Lets replace the values in our formula:

$FV=(1+ \frac{r}{n} )*P[ \frac{(1+ \frac{r}{n})^{kt}-1 }{ \frac{r}{n} } ]$
$FV=(1+ \frac{0.1}{2} )*420[ \frac{(1+ \frac{0.1}{2})^{(2)(15)}-1 }{ \frac{01}{2} } ]$
$FV=29299.53$

We can conclude that the correct answer is $29,299.53