Expert: Josh - 2/16/2008

Question
Hi! So I'm in precalculus and for extra credit our teacher gave us a problem that involves remembering concept that was taught in algebra. Unfortunately, that concept was already forgotten by me.

The situation he created is five different plans that a child is being offered for their weekly allowance. Each plan is different. I have already calculated the different amounts each plan would yield per week for a year, but we were also asked to produce a formula that would give us the total amount for any given week. He hinted that we would have to go back to concepts learned in algebra but I havn't been able to find anything. I faintly remember learning something about formulas, so if you could help me I would really appreciate it! Thanks!

Answer
Hi Amy,

Without the specifics, I can only talk about this in general terms.

I think the idea is to figure out a formula for the amount of money the child has accumulated by week X, where X is a time variable. All these equations will be a function of time, the exact form will depend on the arrangement or payment options.

e.g., Plan 1 may involve a constant weekly allowance of $P. In this case, the amount of money earned by week "t" is given by f(t)=P*t. A straight line curve.

In another situation (I am making this up as I go) the child might be given $30 for the first week of each month, then, the money is halved for each successive week, until the start of the new month. In this case, assuming there are 4 weeks in a month (on average), he/she will receive the sum of 30, 15, 7.5, 3.75 for a given month.

The key is paying attention to how the rate varies with time. For example, if the weekly allowance is increased by 2% each time, this would naturally give rise to a geometric series. Suppose the first week costs $10. The second week's hand out will be $10 + $10*0.02 = 10*(1.02). The third week's will be 2% extra on top of the second week. Continuing this argument, we see the following pattern:
p(1) = 10
p(2) = 10+10*0.02 = 10*1.02
p(3) = (10*1.02)+(10*1.02)*1.02 = 10*(1.02)^2 etc.
The payment for week "t" will be p(t) = 10*(1+0.02)^(t-1)
Applying the geometric sum formula (just do a Google search or look up Wikipedia), the money accumulated by week t is given by f(t)=a*(r^t-1)/(r-1) for r>1. Here, the initial payment "a"=10, the payment ratio between consecutive weeks (viz., p(n+1)/p(n)) is 1.02 by definition. The time variable is given by "t", as in week t.

e.g., you can verify that by week 9, the amount accumulated will be f(t=9) = 10*(1.02^9-1)/(1.02-1) = $97.54. The payments for the first 9 weeks (you can calculate these yourself) are [10.00, 10.20, 10.4040, 10.6121, 10.8243, 11.0408, 11.2616, 11.4869, 11.7166].

I hope you can adapt a similar strategy to the task at hand. Good luck!