Recursive sequences....I'm very confused.

[l3club]

First off, sorry if this is the wrong place to ask this, wasn't sure if I should ask it here or in beginning Algrebra.

Alright, so this is my problem, I'll try to be as clear as possible, but I'm in that state only math can get me in, where I feel ready break something.

So I've just started on recursive sequances, and the site I'm using has this practice problem,

SAVING FOR THE FUTURE:
You are saving for the future.
Your initial deposit is $4100.
Interest is being earned at an annual rate of 5%, compounded monthly.
You will contribute an additional $120 each month.
(a) Find the interest earned in the first month.
(b) Write a recursive formula where u[sub:tp0t97d5]n[/sub:tp0t97d5] gives the amount saved (principal plus interest) after n months.
(c) Then, find the amount saved (principal plus interest) after 7 years.
(d) Find the total amount of money you contributed (principal only) during these 7 years.
(e) Find the total interest earned during these 7 years..

So I did a and then I did b, but I'm having trouble with c. I can't figure out if there's something very simple I'm missing or not. Wouldn't that be u[sub:tp0t97d5]84[/sub:tp0t97d5]? I find it had to believe that I have to work the problem out 84 times, did I not understand something about how I'm supposed to figure the problem out? Or, and I a rather dread this option, am I supposed to be able to do that with my calculator?

For once I'm hoping the answer is I'm just being very dumb and missing something amazingly simple because, though I have a good calculator (my sister gave me her TI-89 graphing calculator) I've been looking at it for the past hour and feel about ready to throw it out the window.

 

[Pklarreich]

I modified the conditions:

SAVING FOR THE FUTURE:
You are saving for the future.
Your initial deposit is $P , which can be p=1200 or anything.
Interest is being earned at an annual rate of R%, compounded monthly, or r = R/12, R can be 5
You will contribute an additional $c each month.
(a) Find the interest earned in the first month.
(b) Write a recursive formula where un gives the amount saved (principal plus interest) after n months.
(c) Then, find the amount saved (principal plus interest) after 7 years.
(d) Find the total amount of money you contributed (principal only) during these 7 years.
(e) Find the total interest earned during these 7 years..

u[0] = P
a) i[1] = Pr
b) u[1] = u[0](1 + r) + c, and in general,
u[k] = u[k-1](1 + r) + c

u[0] = P
u[1] = P(1 + r) + c
u[2] = (P(1 + r) + c)(1 + r) + c
= P(1 + r)^2 + c(1 + r) + c
u[3] = P(1 + r)^3 + c(1 + r)^2 + c(1 + r) + c

u[n] = P(1 + r)^n + c[ (1 + r)^n-1 + ... + 1]
(1 + r)^n - 1
The term in brackets is a geometric series: -------------
1 + r
So that is your formula:

u[n] = P(1 + r)^n + c [(1 + r)^n - 1]/(1 + r)