[Math] Recurrence Relation of Compound Interest

Question

So I have the following question:

Suppose that at the end of some month (referred to as month 0), you open a
investment account with an initial investment of I dollars for some given I .The account grows at a fixed interest rate of r% per month, for a given r (note that if the quoted interest rate is an annual interest, then r = r/12). Suppose that at the end of every month you add some fixed amount m dollars for some given m. Denote x_n by the amount of money you have in your account at the end of
month n.

a) Write a recurrence relation that relates x_n to x_n-1, for n$\ge$t. The relation should involve and m,r,I,n. Make sure to also indicate the value of x₀.

b) Solve the recurrence relation of part (a).

c) Assume that I = \$10,000, r = 0.5%, and m = $1,000. Use part (b) to compute the value of the investment after 20 years

I derived the recurrence relation for part (a) as:

x_n = (1+r)x_n-1 + m      x₀ = I

Then I used the following definition of a linear recurrence relation to solve for part (b)

x_n = a x_n-1 + b ==>  
x_n = (x₀ – ^b⁄_(1-a)
) aⁿ – ^b⁄_(1-a)

Where:

a = (1+r)   &    b = m

So:

x_n = (I – ^m⁄_r) aⁿ – ^m⁄_r

However, when I try to solve for part (c), I get a negative answer (and quite a large one at that).

Where my n = 12 months x 20 years = 240 months

So my assumption is that my initial equation is wrong, but I don't know what else it could be. If anyone has a suggestion of what the proper equation is so I can solve it myself, that would be much appreciated.

Answer 1

The formula you must use is: $$x_n=\left(x_0-\frac{b}{1-a}\right)\cdot a^n \color{red}{+} \frac{b}{1-a}.$$