I can't seem to come up with an equation.

Question

This problem seems a lot simpler than I think it is. It seems to have an element of both a geometric sequence (by the constant increase of 5%) and also arithmetic (with the inclusion of the $2 fee every month). I keep getting close to the actual answer (which is $78.20) but off by about a few dollars.

 

Charles lost two rented movies, so he owes the rental store a fee of $40. At the end of each month, the amount that Charles owes will increase by 5%, plus a $2 billing fee. Define a recursive and explicit function. How much money will Charles owe the rental store after 8 months?

Answer 1

Charles ought to pay up ASAP in order to stop the accumulation of ever larger bills!

 

Let's say that now he owes X₁ =$40. The next month he owes 1.05(40) + 2 = 44 = X₂.

 

Continuing by using recursion: x_n+1 = x_n(1.05) + 2 for n ≥ 1

 

 

 

Answer 2

a₀ = 40

r = 1.05

d = 2

a₁ = ra₀ + d

a₂ = ra₁ + d

    = r[(ra₀ + d] + d

    = r²a₀ + d(1 + r)

a₃ = ra₂ + d

    = r[^r2a₀ + d(1 + r)] + d

    = r³a₀ + d[1 + r + r²]

    = r³a₀ + d[(r³ -1)/(r -1)]

a_n = ra_n-1 + d

    = rⁿa₀ + d[(rⁿ -1)/(r -1)]

[Explicit Formula]

 

After 8 months

a₈ = r⁸a₀ + d[(r⁸ -1)/(r -1)]

    = 1.05⁸(40) + 2 [(1.05⁸ - 1)/(1.05 - 1)]

    = 78.20