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?
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
"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
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