Answer to Question #318612 in Financial Math for Zia

Question

You owe $500 on a credit card that charges 1.5% interest each month. You can pay $50 each month with no new charges. What is the equilibrium value? What does the equilibrium value mean in terms of the credit card? Build a numerical solution. When will the account be paid off? How much is the last payment?

Answer 1

Let a_n be the amount on a credit card that charges 1.5% interest each month. Suppose, a person pay $50 each month, then the dynamical system as follows:

The equilibrium value for the dynamical system is

$a_{n+1}=ra_n+b, \ r \not=1$

Here, $a_{n+1}=a_n+0.015a_n-50$

whith $a_0=500$

For, $n=0$

$a_{1}=a_0+0.015a_0-50$

$a_{1}=500+0.015(500)-50=457.5$

For n=1

$a_{2}=a_1+0.015a_1-50$

$a_{2}=457.5+0.015(457.5)-50=414.3625$

$a_{3}=a_2+0.015a_2-50$

$a_{3}=414.3625+0.015(414.3625)-50=370.5779$

$a_{4}=a_3+0.015a_3-50$

$a_{4}=370.5779+0.015(370.5778)-50=326.1366$

$a_{5}=a_4+0.015a_4-50$

$a_{5}=326.1366+0.015(326.1366)-50=281.0287$

$a_{6}=a_5+0.015a_5-50$

$a_{6}=281.0287+0.015(281.0281)-50=235,2441$

$a_{7}=a_6+0.015a_6-50$

$a_{7}=235,2441+0.015(235,2441)-50=188.7727$

$a_{8}=a_7+0.015a_7-50$

$a_{8}=188.7727+0.015(188.7727)-50=141.6043$

$a_{9}=a_8+0.015a_8-50$

$a_{9}=141.6043+0.015(141.6043)-50=93.7284$

$a_{10}=a_9+0.015a_9-50$

$a_{10}=93.7284+0.015(93.7284)-50=45.1343$

$a_{11}=a_{10}+0.015a_{10}-50$

$a_{11}=45.1343+0.015(45.1343)-50=-4.1887$

Account will be paid off for 11 months.

Last payment will be 50-4.19=45.81