A rich uncle has offered to pay you 1000 on Sept 1 If you go

A rich uncle has offered to pay you $1000 on Sept 1 If you go to college. As long as you stay In school that semester, he also promises to pay you 9/l0^th of the $1000 on Sept. 2, and 9/10^th of the Sept. 2nd amount on Sept. 3, etc. You are done with finals on Dec. 15, the last day he will pay you. Determine how much he will pay you on October 15. (round to the nearest penny) Determine how much he will pay you on December 15. (round to the nearest penny) If you do not get a GPA of 3.0 or higher, your uncle wants his money back. How much would you have to repay? (round to the nearest penny)

Solution

This is a progresion, and you will use the following expression:
An = A1*r^(n-1)

Where n would be the number of days, so

A2 = 1000 * (9/10)¹ = 900$
A3 = 1000 * (9/10)² = 810$
A4 = 1000 * (9/10)³ = 729$

And so on. For october 15 it\'s been 45 days so:
A45 = 1000*(9/10)⁴⁵ = 8.73 $

For December 15th it\'s been 3 and a half month, if we round to 30 days per month, we have 30 of september, 30 october, 30 november and 15 december. That\'s 105 days (or 106 if you take account the 31 days of october) so:
A₁₀₆ = 1000 * (9/10)¹⁰⁶ = 0.01 cents.

To payback is the sum of all days:
Sn = A1 * (rⁿ-1/r-1)
S₁₀₆ = 1000 * (9/10)¹⁰⁶ - 1 / (9/10)-1
S₁₀₆ = 9999.86 $

Hope this helps