864 CHAPTER 12 Sequences; Induction; the Binomial Theorem Recursive sequences can also be used to compute information about loans.When equal periodic payments are made to pay off a loan, the loan is said to be amortized . Mortgage Payments John and Wanda borrowed $180,000 at 7% per annum compounded monthly for 30 years to purchase a home. Their monthly payment is determined to be $1197.54. (a) Find a recursive formula that represents their balance after each payment of $1197.54 has been made. (b) Determine their balance after the first payment is made. (c) When will their balance be below $170,000? EXAMPLE 11 Solution (a) Use formula (10) with = = A r 180,000, 0.07, 0 and = P $1197.54. Then ( ) = = + − − A A A 180,000 1 0.07 12 1197.54 n n 0 1 (b) In SEQuence mode on a TI-84 Plus CE, enter the sequence { } An and create Table 6. After the first payment is made, the balance is = A $179,852. 1 (c) Scroll down until the balance is below $170,000. See Table 7. After the fifty-eighth payment is made ( ) = n 58 , or 4 years, 10 months, the balance is below $170,000. (c) On March 1 ( ) = n 6 , there is only $1107.53, not enough to pay for the trip to Cancun. (d) If the periodic deposit, P, is $125, then on March 1 n 6 ( ) = , there is $1258.15 in the account, enough for the trip. See Table 5. Now Work PROBLEM 85 THEOREM Amortization Formula If B$ is borrowed at an interest rate of r% (expressed as a decimal) per annum compounded monthly, the balance An due after n monthly payments of P$ is given by the recursive sequence ( ) = = + − ≥ − A B A r A P n 1 12 1 n n 0 1 (10) Table 7 Table 6 Formula (10) may be explained as follows: The initial loan balance is B$ . The balance due An after n payments will equal the balance due previously, − A , n 1 plus the interest charged on that amount, reduced by the periodic payment P. Now Work PROBLEM 87 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 12.1 Assess Your Understanding 1. For the function ( ) = − f x x x 1 , find ( ) f 2 and ( ) f 3 . (pp. 61–68) 2. True or False A function is a relation between two sets D and R so that each element x in the first set D is related to exactly one element y in the second set R. (pp. 61–68) 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure Table 5
RkJQdWJsaXNoZXIy NjM5ODQ=