+123 Luke is borrowing $10000 from the bank. The bank offers him a choice between two 10-year payment plans:
Plan 1: Luke's debt accumulates 10% annual interest which compounds quarterly. Luke pays off half his balance after 5 years, and the rest at the end of the 10 years.
Plan2: Luke's debt accumulates 10% annual interest which compounds annually. Luke pays off his full balance at the end of the 10 years.
What is the (positive) difference between Luke's total payments under Plan 1 and his total payments under Plan 2? Round to the nearest dollar.
+6251 \(r=0.1\\ n_1=4,~n_2=1\\ p=10000\\ \text{plan 1}:\\ payment_{1,1} =\dfrac 1 2 p\left(1+\dfrac{r}{n_1}\right)^{5n_1}\\ payment_{1,2} = payment_{1,1}\left(1+\dfrac{r}{n_1}\right)^{5n_1}\\ p_1 = \text{Plan 1 total payment = }payment_{1,1} + payment_{1,2}\)
\(\text{Plan 2}:\\ payment_{2,1} = p\left(1+\dfrac{r}{n_2}\right)^{10 n_2}\\ p_2=\text{Plan 2 total payment = }payment_{2,1}\\ \text{find }|p_2-p_1|\)
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