Michelle is attending college and has a part-time job. once she finishes college, michelle would like to relocate to a metropolitan area. she wants to build her savings so that she will have a "nest egg" to start her off. michelle works out her budget and decides she can afford to set aside $130130 per month for savings. her bank will pay her 5 %5% per year, compounded monthly, on her savings account. what will be michelle's balance in five years?

$8877.62 This is an example of an investment with compound interest and regular contributions each period. So let's examine it closely. To make things easier, I'll work with individual months instead of entire years. So we'll have 60 periods (12 * 5) and the interest rate for each period will be 5/12 = 0.41666%. For the interest, we'll add 1 and call the value z, so that our math doesn't have to keep adding. The contribution each month will be called C which will be made at the beginning of the month. So let's look at the end of the first few months. 1. Cz 2. Cz^2 + Cz = C(z+ z^2) 3. Cz^3 + Cz^2 + Cz = C(z + z^2 + z^3) ... n. Cz^n + Cz^(n-1) + ... + Cz = C(z + z^2 + ... + z^n) The (z + z^2 + ... + z^n) part is the sum of a geometric series. If you look it up, you'll see that it reduces to this (z^(n+1) - z)/(z-1) Now we can calculate how much Michelle will have after 60 months. T = 130(z^61 - z)/(z-1) z = 1 + 0.05/12 = 1.004166667 so T = 130(1.004166667^61 - 1.004166667)/(1.004166667-1) T = 130(1.288706006 - 1.004166667)/.004166667 T = 130(0.28453934)/.004166667 T = 130(68.28944152) T = 8877.627398 So Michelle will have after 5 years, $8877.62 saved.

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