needOfHelpCMath said:
In 6 years, Mattos Oil Refining Company wants to pay off a $50,000 debt in one lump sum amount. They must set up an account to accumulate the necessary funds to pay off their debt. If the payments are made every 6 months and the fund earns 9.6% compounded semiannually, what is the amount of each semiannual payment?
I have used this formula cannot get the answer? what is problem? Is this the correct formula
Periodic Deposit for Annuity or Sinking Fund
R = S[(r/m)/((1+r/m)^(mt)-1)]
Well, let's see if we can come up with a formula... It compounds semi-annually, so let's call the semi-annual interest rate $\displaystyle \begin{align*} R \end{align*}$. Each half year a payment of $\displaystyle \begin{align*} D \end{align*}$ is added. Thus...
$\displaystyle \begin{align*} V_1 &= V_0 \left( 1 + R \right) + D \\ \\ V_2 &= V_1 \left( 1 + R \right) + D \\ &= \left[ V_0 \left( 1 + R \right) + D \right] \left( 1 + R \right) + D \\ &= V_0 \left( 1 + R \right) ^2 + \left[ D \left( 1 + R \right) + D \right] \\ \\ V_3 &= V_2 \left( 1 + R \right) + D \\ &= \left[ V_0 \left( 1 + R \right) ^2 + D \left( 1 + R \right) + D \right] \left( 1 + R \right) + D \\ &= V_0 \left( 1 + R \right) ^3 + \left[ D \left( 1 + R \right) ^2 + D \left( 1 + R \right) + D \right] \\ \\ V_4 &= V_3 \left( 1 + R \right) + D \\ &= \left[ V_0 \left( 1 + R \right) ^3 + D \left( 1 + R \right) ^2 + D \left( 1 + R \right) + D \right] \left( 1 + R \right) + D \\ &= V_0 \left( 1 + R \right) ^4 + \left[ D \left( 1 + R \right) ^3 + D \left( 1 + R \right) ^2 + D \left( 1 + R \right) + D \right] \end{align*}$
So the pattern appears to be...
$\displaystyle \begin{align*} V_n &= V_0 \left( 1 + R \right) ^n + \left[ D \left( 1 + R \right) ^{n - 1} + D \left( 1 + R \right) ^{n - 2} + D \left( 1 + R \right) ^{n - 3} + \dots + D \left( 1 + R \right) ^2 + D \left( 1 + R \right) + D \right] \end{align*}$
and since the terms involving D form a geometric series $\displaystyle \begin{align*} S_n = a + a\,r + a\,r^2 + a\,r^3 + \dots + a\,r^{n - 1} \end{align*}$, it can be written in a closed form as $\displaystyle \begin{align*} \frac{a \left( r^n - 1 \right) }{r - 1} \end{align*}$ giving
$\displaystyle \begin{align*} V_n &= V_0 \left( 1 + R \right) ^n + \frac{ D \left[ \left( 1 + R \right) ^n - 1 \right] }{ \left( 1 + R \right) - 1 } \\ V_n &= V_0 \left( 1 + R \right) ^n + \frac{D \left[ \left( 1 + R \right) ^{n} - 1 \right] }{R} \end{align*}$Now in your case, you haven't listed all the necessary information, and what you have posted is ambiguous. I am ASSUMING that the 9.6% interest rate is the interest rate per annum, so that means that your half yearly interest rate is $\displaystyle \begin{align*} R = 4.8\% = 0.048 \end{align*}$. You haven't said what your initial investment $\displaystyle \begin{align*} V_0 \end{align*}$ is. You need to make 12 payments (one every half year for 6 years) so $\displaystyle \begin{align*} n = 12 \end{align*}$ and you need to amount to $\displaystyle \begin{align*} V_{12} = 50\,000 \end{align*}$. So with the $\displaystyle \begin{align*} V_0 \end{align*}$ value you should have, you substitute all these values in and then solve for $\displaystyle \begin{align*} D \end{align*}$.
