Answer to Question #113376 in Algebra for Rosi

Question

Question 1:
A couple decides to buy a house which is currently valued at $318,921.46 on loan. The couple is willing to start paying $200.00 per month and are willing to increase their payment at a rate of 5% every month. How many payments are necessary to pay off the loan amount assuming no deposit was made (answer to the nearest whole number)? What is the value of the final payment that they would make assuming no deposit was made (answer to the nearest whole number)? (hint:use geometric sequence)

Answer 1

$b_1=200\\ q=1+\frac{5}{100}=1.05\\ S_n=318921.46\\ S_n=\frac{b_1(1-q^n)}{1-q}\\ \frac{200(1-1.05^n)}{1-1.05}=318921.46\\ 1-1.05^n=-79.73\\ 1.05^n=80.73\\ n=\log_{1.05}80.73\\ n=90$

90 payments are necessary to pay off the loan amount assuming no deposit was made

$b_n=b_1q^{n-1}\\ b_{90}=200\cdot1.05^{89}\\ b_{90}=15378$

15378 is the value of the final payment that they would make assuming no deposit was made