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LOANS AND INVESTMENTS
Jump right to the exercises!
Loan and investment problems offer great applications of recursive sequences.
PAYING BACK A LOAN:
Suppose you are borrowing $22,000.
Interest is being charged at an annual rate of 5%, compounded monthly.
You plan to pay back $250 each month; this payment goes to both interest and principal.
(a) Find the interest owed in the first month.
(b) Write a recursive formula where un
gives the amount owed after n months.
(c) Then, find the amount owed after one year of payback.
(d) Find the total principal paid in the first year.
(e) Find the total interest paid in the first year.
SOLUTION:
(a) The interest owed in the first month is ($22,000)(0.0512) = $91.67 .
(b) The recursive formula is:
u0=22000 ;
un=(1
+0.0512)
un-1
-250
for n≥1 .
(c)
(Note: For this part of the problem, you should have a calculator that does recursion.)
From the calculator,
u12=20055.85 .
Thus, you owe $20,055.85 after one year.
(d) The total principal paid in the first year is $22,000 - $20,055.85 = $1944.15 .
(e) The total interest paid in the first year is (12)($250) - $1944.15 = $1055.85 .
SAVING FOR THE FUTURE:
You are saving for the future.
Your initial deposit is $4100.
Interest is being earned at an annual rate of 5%, compounded monthly.
You will contribute an additional $120 each month.
(a) Find the interest earned in the first month.
(b) Write a recursive formula where un
gives the amount saved (principal plus interest) after n months.
(c) Then, find the amount saved (principal plus interest) after 7 years.
(d) Find the total amount of money you contributed (principal only) during these 7 years.
(e) Find the total interest earned during these 7 years.
SOLUTION:
(a) The interest earned in the first month is ($4,100)(0.0512) = $17.08 .
(b) The recursive formula is:
u0=4100 ;
un=(1
+0.0512)
un-1
+120
for n≥1 .
(c) From the calculator,
u84=17853.39 .
Thus, you have saved (principal plus interest) $17,853.39 after 7 years.
(d) The total amount of money you contributed (principal only) during these 7 years is $4,100 + 7(12)($120) = $14,180.00 .
(e) The total interest earned during these 7 years is $17,853.39 - $14,180.00 = $3,673.39 .
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.