Sequences Help

[dhthompson]

Phil bought a car by taking out a loan for $24,500 at 0.5% interest per month. Phil's normal monthly payment is $534.47 per month, but he decides that he can afford to pay $150 extra toward the balance each month. Write a recursively defined sequence for the balance he has each month.

 

[chrisr]

$\displaystyle He\ decides\ to\ make\ monthly\ repayments\ of\ 684.47\ dollars.$

$\displaystyle Balance\ after\ first\ month\ is\ 24,500(1.005)-684.47\ dollars.$

$\displaystyle Balance\ after\ second\ month\ is\ [24,500(1.005)-684.47](1.005)-684.47\ dollars$
$\displaystyle =(previous\ month's\ balance)(1.005)-684.47\ dollars$
$\displaystyle =24,500(1.005)^2-684.47(1.005)-684.47\ dollars.$

$\displaystyle Balance\ after\ third\ month\ is\ [24,500(1.005)^2-684.47(1.005)-684.47](1.005)-684.47$
$\displaystyle =(previous\ month's\ balance)(1.005)-684.47\ dollars.$
$\displaystyle =24,500(1.005)^3-684.47(1.005)^2-684.47(1.005)-684.47\ dollars.$

$\displaystyle Try\ formulating\ from\ here.$

 

[Denis]

or:

a = 24500, p = 684.47, i = .005, n = number of months

After n months:

a(1+i)^n - p[(1+i)^n - 1] / i