Modeling problems with recurrence relation

Question

A bank adds 5% interest to savings accounts at the end of every year. Saver deposits 2500 at the beginning of a certain year. Then at the end of every year the saver withdraws 100. Let A_n be the balance after n years.

A- find a recurrence relation, together with an initial condition.

B- use the recurrence relation and initial condition to compute the balance after 4 years.

I worked A a little and I've got An_1=1000-420 but I don't know if this is correct because I'm not getting the formula right.

Answer 1

Your answer for the value at the end of year 1 isn't quite right. First you add the interest on the balance, then subtract the withdrawal, so $$ A_1 = (1.05) \times 2500 - 100. $$ If you think of that $2500$ as $A_0$ you should now be able to write $A_n$ in terms of $A_{n-1}$ and finish the problem.

Answer 2

Hint: The recurrence relation is supposed to be of the form $A_n=$ some function of $A_{n-1}$. If you have $A_{n-1}$ at the end of year $n-1$ you get some interest and lose a withdrawal by the end of year $n$. It seems, but is not completely clear from the problem statement, that the interest is calculated on the balance before the withdrawal. How much is the interest? Can you write the recurrence relation?