Question #318612

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?


1
Expert's answer
2022-03-29T06:48:28-0400

Let an 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

an+1=ran+b, r1a_{n+1}=ra_n+b, \ r \not=1

Here, an+1=an+0.015an50a_{n+1}=a_n+0.015a_n-50

whith a0=500a_0=500

For, n=0n=0

a1=a0+0.015a050a_{1}=a_0+0.015a_0-50

a1=500+0.015(500)50=457.5a_{1}=500+0.015(500)-50=457.5

For n=1

a2=a1+0.015a150a_{2}=a_1+0.015a_1-50

a2=457.5+0.015(457.5)50=414.3625a_{2}=457.5+0.015(457.5)-50=414.3625

a3=a2+0.015a250a_{3}=a_2+0.015a_2-50

a3=414.3625+0.015(414.3625)50=370.5779a_{3}=414.3625+0.015(414.3625)-50=370.5779

a4=a3+0.015a350a_{4}=a_3+0.015a_3-50

a4=370.5779+0.015(370.5778)50=326.1366a_{4}=370.5779+0.015(370.5778)-50=326.1366

a5=a4+0.015a450a_{5}=a_4+0.015a_4-50

a5=326.1366+0.015(326.1366)50=281.0287a_{5}=326.1366+0.015(326.1366)-50=281.0287

a6=a5+0.015a550a_{6}=a_5+0.015a_5-50

a6=281.0287+0.015(281.0281)50=235,2441a_{6}=281.0287+0.015(281.0281)-50=235,2441

a7=a6+0.015a650a_{7}=a_6+0.015a_6-50

a7=235,2441+0.015(235,2441)50=188.7727a_{7}=235,2441+0.015(235,2441)-50=188.7727

a8=a7+0.015a750a_{8}=a_7+0.015a_7-50

a8=188.7727+0.015(188.7727)50=141.6043a_{8}=188.7727+0.015(188.7727)-50=141.6043

a9=a8+0.015a850a_{9}=a_8+0.015a_8-50

a9=141.6043+0.015(141.6043)50=93.7284a_{9}=141.6043+0.015(141.6043)-50=93.7284

a10=a9+0.015a950a_{10}=a_9+0.015a_9-50

a10=93.7284+0.015(93.7284)50=45.1343a_{10}=93.7284+0.015(93.7284)-50=45.1343

a11=a10+0.015a1050a_{11}=a_{10}+0.015a_{10}-50

a11=45.1343+0.015(45.1343)50=4.1887a_{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




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