find the payment that should be used for the annuity due whose future value is given. assume that the…

find the payment that should be used for the annuity due whose future value is given. assume that the compounding period is the same as the payment period. $17,000, quarterly payments for 12 years; interest rate 5.6%. the payment should be $ (do not round until the final answer. then round to the nearest cent as needed.)

find the payment that should be used for the annuity due whose future value is given. assume that the compounding period is the same as the payment period. $17,000, quarterly payments for 12 years; interest rate 5.6%. the payment should be $ (do not round until the final answer. then round to the nearest cent as needed.)

Answer

Explanation:

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

The formula for the future - value of an annuity due is $F = A\times\frac{(1 + r)^{n}-1}{r}\times(1 + r)$, where $F$ is the future value of the annuity, $A$ is the payment per period, $r$ is the interest rate per period, and $n$ is the number of periods. We need to solve for $A$. Rearranging the formula for $A$ gives $A=\frac{F\times r}{(1 + r)^{n}-1}\times\frac{1}{1 + r}$.

Step2: Determine the values of $F$, $r$, and $n$

The future value $F=$17000$. The annual interest rate $i = 5.6%=0.056$. Since the payments are quarterly, the interest rate per period $r=\frac{0.056}{4}=0.014$. The number of years is 12 years, and since there are 4 quarters in a year, the number of periods $n = 12\times4=48$.

Step3: Substitute the values into the formula

$A=\frac{17000\times0.014}{(1 + 0.014)^{48}-1}\times\frac{1}{1 + 0.014}$ First, calculate $(1 + 0.014)^{48}$. Using the formula $a^{b}$, where $a = 1.014$ and $b = 48$, we have $(1.014)^{48}\approx1.95797$. Then, $(1.014)^{48}-1\approx1.95797 - 1=0.95797$. $17000\times0.014 = 238$. $\frac{238}{0.95797}\approx248.44$. $\frac{248.44}{1.014}\approx245.01$.

Answer:

$245.01$