Discrete Dynamical Systems & Credit Card Debt: How to solve for payment

I have the following problem, taken out of Giordano, Fox, and Horton's A First Course in Mathematical Modeling:

Your current credit card balance is $\$12,000$with a current rate of$19.9\%$per year. Interest is charged monthly. Determine what monthly payment p will pay off the card in two years, and in four years. Now, assume that each month you charge$\$105$. Determine what monthly payment $p$ will pay off the card in two years, and in four years.

This problem wasn't too tricky, but I got stuck toward the end, when assuming that at time $t=0$, payment $p=0$. Allow me to show my work:

Let $a_t$ denote the amount of the debt at time $t$. We are given that $a_0 = \$12,000.$Since interest is charged monthly, we know that the interest rate is$i = \frac{0.199}{12} = 0.0165833333...$. Since we are charging an additional$\$105$ per month, and intend to make payment $p$ to pay everything off, our balance can be modeled accordingly: $$a_{t+1} = a_t(1+i)+105-p$$ $$a_{t+1} = 1.01658333...\cdot a_t+105-p$$ The solution to this equation will take on the form: $$a_t = 1.01658333...^t \cdot c - \frac{105-p}{1-1.01658333...}$$ $$a_t = 1.01658333...^t \cdot c + \frac{105-p}{0.01658333...}$$ for some scalar $c$. To find the value of $c$, we will suppose that at time $t=0$, no payments would be made:

$$a_0 = 1.01658333...^0 \cdot c + \frac{105-0}{0.01658333...}$$ $$12000= c + \frac{105}{0.01658333...}$$ $$\implies c \approx 5668.3417085.$$

My question is this: was I right to assume that at $t=0, \space p=0$? Otherwise, I'm not sure how else to find $c$.

Any help or insight would be greatly appreciated!

No, $p$ is a constant by assumption. Two simple thoughts: the additional $105$ charged each month just has to be added to the payment to pay off the loan if you don't charge any more, so you can solve the problem without the $105$ and add it in at the end. Don't be so quick to put in the value of $i$. You will save yourself a lot of typing and at the end have a formula that you can use for other values of $i$. You might see problems like this again.
Your first line under the solution to this equation is wrong. Increasing $p$ will increase $a_t$, which cannot be right. What you should do is write the equation you did for $a_0$ but without setting $p=0$. Then write the equation $a_{24}=0$ for the two year term. That gives you two equations in two unknowns $c,p$ that you can solve. For the four year term, change it to $a_{48}=0$
• Sorry, the first line under the solution should be: $$a_t = (1+i)^t \cdot c + \frac{105-p}{-i},$$ correct? – daOnlyBG Feb 24 '15 at 4:45
• Thanks much. I don't know why it didn't occur to me that I have two points in the form $(t, a_t)$ to take advantage of. I believe it should be smooth sailing from here, and yes, perhaps it is better to not rush the $i$ substitution. – daOnlyBG Feb 24 '15 at 5:17