Chapter 6 Finance. 5.5 Annuities and Amortization Using Recursive Sequences

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Hester Ramsey
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1 Chapter 6 Finance 5.5 Annuities and Amortization Using Recursive Sequences 3. Credit Card Debt John has a balance of $3000 on his credit card that charges 1% interest per month on any unpaid balance. John can afford to pay $100 toward the balance each month. His balance each month after making a $100 payment is given by the recursively defined sequence B 0 = $3000, Bn = 1.01Bn (b) Using a graphing utility, determine when John s balance will be below $2000. How many of the $100 payments have been made? Before we can enter the information for the sequence, we must first put the calculator in sequence mode. z ~ ~ ~ Í Enter the recursive definition of the sequence in the function editor. The calculator uses n for the variable and the letters u, v, and w for the names of the sequences. Note that the variable n is found on the key, and the letters u, v, and w are above the,, and keys, respectively. o } Ê Í À Ë Ê À y ¹ À ¹ À Ê Ê Í Â Ê Ê Ê Í 57

2 Next we must set up the table feature so we can choose the values for n. y p ~ Í Now, create the table. y s Ê Í À Í Á Í Â Í Í Í Í Continue to enter values. Be sure to move back the top of the first column. } } } } } } Í Í Í À Ê Í À À Í À Á Í À Â Í 58

3 Continue to enter values. After the 14 th payment, the balance is below $2000. (c) Using a graphing utility, determine when John will pay off the balance. What is the total of all of the payments? Continue to enter values in the table. You can use the { key to delete unwanted entries from the table. John will have to make 36 payments. John s final payment should be $84.62 (since he overpaid by $15.38), so John s total of all payments is $ $84.62 = $3, Be sure to set your calculator back in Func (function) mode before you try to graph any functions. To set your calculator in Func mode, press z, move the cursor down to Func and press Í. 59

4 5. Trout Population A pond currently contains 2000 trout. A fish hatchery decides to add an additional 20 trout each month. In addition, it is known that the trout population is growing 3% per month. The size of the population after n months is given by the recursively defined sequence p 0 = 2000, pn = 1.03pn (b) Using a graphing utility, determine how long it will be before the trout population reaches Enter the recursive definition of the sequence in the function editor. Using the table feature, enter values for n until the value of the sequence reaches The population will reach 5000 in the 26 th month. Be sure to set your calculator back in Func (function) mode before you try to graph any functions. To set your calculator in Func mode, press z, move the cursor down to Func and press Í. 60

5 7. Roth IRA On January 1, 2015, Bob decides to place $500 at the end of each quarter into a Roth Individual Retirement Account. (b) How long will it be before the value of the account exceeds $100,000? Enter the recursive definition of the sequence in the function editor. Using the table feature, enter values for n. until you pass After the 82 nd payment. (c) What will be the value of the account in 25 years when Bob retires? When Bob retires, he will have made 100 payments. The value is $156,

6 9. Home Loan Bill and Laura borrowed $150,000 at 6% per annum compounded monthly for 30 years to purchase a home. Their monthly payment is determined to be $ (c) Using a graphing utility, create a table showing Bill and Laura s balance after each monthly payment. Enter the recursive definition of the sequence in the function editor. Using the table feature, enter values for n. To complete the table, enter values from 21 to 360 for n. (d) Using a graphing utility, determine when Bill and Laura s balance will be below $140,

7 As you continue to enter values for n you will obtain the following. After the 58 th payment. (e) Using a graphing utility, determine when Bill and Laura will pay off the balance. As you continue to enter values for n you will obtain the following. After the 360 th payment. (g) Suppose that Bill and Laura decide to pay an additional $100 each month on their loan. Answer parts (a) to (f) under this scenario. (c) Using a graphing utility, create a table showing Bill and Laura s balance after each monthly payment. 63

8 To complete the table, enter values from 21 on for n. (d) Using a graphing utility, determine when Bill and Laura s balance will be below $140,000. As you continue to enter values for n you will obtain the following. After the 37 th payment. (e) Using a graphing utility, determine when Bill and Laura will pay off the balance. As you continue to enter values for n you will obtain the following. After the 279 th payment. 64

9 (h) Is it worthwhile for Bill and Laura to pay the additional $100. Explain. Yes. If Bill and Laura make a monthly payment of $899.33, then the total interest paid is $173, If Bill and Laura make a monthly payment of $999.33, then the total interest paid is $128, Bill and Laura saved $45, in interest payments. 65

10 Summary The command introduced in this chapter is: n x x 66

The values in the TVM Solver are quantities involved in compound interest and annuities.

The values in the TVM Solver are quantities involved in compound interest and annuities. Texas Instruments Graphing Calculators have a built in app that may be used to compute quantities involved in compound interest, annuities, and amortization. For the examples below, we ll utilize the screens