Math Struggles: Geometric Series & Paying Off a $200 Balance

[Calixto]

What the heck?

The minimum monthly payment for a credit card is the larger of $5 or 1/25 of the outstanding balance. If the balance is less than $5, then the entire balance is due. If you make only the minimum payment each month, how long will it take to pay off a balance of $200?


Clearly, this has to do with geometric series. I can conceptually understand this problem, but I'm having trouble putting it into mathematical terms while relating to geometric series. If you have any advice, I would greatly appreciate it. Thanks.

 

[HallsofIvy]

First, when do you start making a minimum payment of $5? x/25= 5 when x= 125 so how many payments will be required to bring the balance down to $125? If your initial balance is S and you pay a fraction r of that each month, you first payment will be rS and the remaining balance S- rS= S(1-r). Your second payment will be r(1-r)S and the remaining balance then will be S(1-r)- r(1-r)S= (1-r)(S- rS)= (1-r)²S.

The remaining balance, after n payments, is the geometric sequence (1-r)ⁿS. For what n is (1- 1/25)ⁿ(200)< 125? At that point the balance will be between 120 and 125 and will require 120/5= 22 payments of $5 each and a final payment of less than $5.

 

[tiny-tim]

What the heck?

oh … I have never seen such language!

Tush! And pish!

Clearly, this has to do with geometric series. I can conceptually understand this problem, but I'm having trouble putting it into mathematical terms while relating to geometric series. If you have any advice, I would greatly appreciate it. Thanks.

Hi Calixto!

Geometric sequence, actually.

General advice:

Be systematic.

Choose a name, like Pn, for the amount of money remaining after n months, and then find the formula connecting Pn and Pn-1.

 

[Calixto]

Ok thanks HallsOfIvy, that helps a lot. And sorry tiny-tim for using such offensive language.