SECTION 6.1 ✓ CHECKPOINT ANSWERS Simple Interest; Sequences 367 I ϭ Prt S ϭ $8360 r ϭ 0.12, or 12% Only (b) is an arithmetic sequence; the common difference is (a) a51 ϭ 190 (b) s51 ϭ 4590 | EXERCISES | 6.1 SIMPLE INTEREST In Problems 1–4, find the requested value and tell what the other numbers represent Find r: 250 ϭ 1000(r)(4) Find t: 1500 ϭ 5000(0.05)(t) Find P: 9600 ϭ P ϩ P(0.05)(4) Find P: 11,800 ϭ P ϩ P(0.03)(6) $10,000 is invested for years at an annual simple interest rate of 16% (a) How much interest will be earned? (b) What is the future value of the investment at the end of the years? $800 is invested for years at an annual simple interest rate of 14% (a) How much interest will be earned? (b) What is the future value of the investment at the end of the years? $1000 is invested for months at an annual simple interest rate of 12% (a) How much interest will be earned? (b) What is the future value of the investment after 3 months? $1800 is invested for months at an annual simple interest rate of 15% (a) How much interest will be earned? (b) What is the future value of the investment after 9 months? If you borrow $800 for months at 16% annual simple interest, how much must you repay at the end of the 6 months? 10 If you borrow $1600 for years at 14% annual simple interest, how much must you repay at the end of the 2 years? 11 If you lend $3500 to a friend for 15 months at 8% annual simple interest, find the future value of the loan 12 Mrs Gonzalez lent $2500 to her son Luis for months at 9% annual simple interest What is the future value of this loan? 13 A couple bought some stock for $30 per share that pays an annual dividend of $0.90 per share After year the price of the stock was $33 Find the simple interest rate on the growth of their investment 14 Jenny Reed bought SSX stock for $16 per share The annual dividend was $1.50 per share, and after year SSX was selling for $35 per share Find the simple interest rate of growth of her money 15 (a) To buy a Treasury bill (T-bill) that matures to $10,000 in months, you must pay $9750 What annual simple interest rate does this earn? (b) If the bank charges a fee of $40 to buy a T-bill, what is the actual interest rate you earn? 16 Janie Christopher lent $6000 to a friend for 90 days at 12% After 30 days, she sold the note to a third party for $6000 What annual simple interest rate did the third party receive? Use 360 days in a year 17 A firm buys 12 file cabinets at $140 each, with the bill due in 90 days How much must the firm deposit now to have enough to pay the bill if money is worth 12% simple interest per year? Use 360 days in a year 18 A student has a savings account earning 9% simple interest She must pay $1500 for first-semester tuition by September and $1500 for second-semester tuition by January How much must she earn in the summer (by September 1) in order to pay the first-semester bill on time and still have the remainder of her summer earnings grow to $1500 between September and January 1? 19 If you want to earn 15% annual simple interest on an investment, how much should you pay for a note that will be worth $13,500 in 10 months? 20 What is the present value of an investment at 6% annual simple interest if it is worth $832 in months? 21 If $5000 is invested at 8% annual simple interest, how long does it take to be worth $9000? 22 How long does it take for $8500 invested at 11% annual simple interest to be worth $13,000? 23 A retailer owes a wholesaler $500,000 due in 45 days If the payment is 15 days late, there is a 1% penalty charge The retailer can get a 45-day certificate of deposit (CD) paying 6% or a 60-day certificate paying 7% Is it better to take the 45-day certificate and pay on time or to take the 60-day certificate and pay late with the penalty? 24 An investor owns several apartment buildings The taxes on these buildings total $30,000 per year and are due before April The late fee is 2% per month up to months, at which time the buildings are seized by the authorities and sold for back taxes If the investor has $30,000 available on March 31, will he save money by paying the taxes at that time or by investing the money at 8% and paying the taxes and the penalty on September 30? Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 368 CHAPTER Mathematics of Finance 25 Bill Casler bought a $2000, 9-month certificate of deposit (CD) that would earn 8% annual simple interest Three months before the CD was due to mature, Bill needed his CD money, so a friend agreed to lend him money and receive the value of the CD when it matured (a) What is the value of the CD when it matures? (b) If their agreement allowed the friend to earn a 10% annual simple interest return on his loan to Bill, how much did Bill receive from his friend? 26 Suppose you lent $5000 to friend for 18 months at an annual simple interest rate of 9% After year you need money for an emergency and decide to sell the note to friend (a) How much does friend owe when the loan is due? (b) If your agreement with friend means she earns simple interest at an annual rate of 12%, how much did friend pay you for the note? 43 Find the sum of the first 70 terms of the arithmetic sequence with first term 10 and common difference 21 44 Find the sum of the first 80 terms of the arithmetic sequence with first term 12 and common difference Ϫ3 45 Find the sum of the first 150 terms of the arithmetic sequence 6, 92, 3, 46 Find the sum of the first 200 terms of the arithmetic sequence 12, 9, 6, APPLICATIONS 47 Bee reproduction A female bee hatches from a fertilized egg, whereas a male bee hatches from an unfertilized egg Thus a female bee has a male parent and a female parent, but a male bee has only a female parent Therefore, the number of ancestors of a male bee follows the Fibonacci sequence 1, 2, 3, 5, 8, 13, … SEQUENCES 27 Write the first ten terms of the sequence defined by an ϭ 3n 28 Write the first seven terms of the sequence defined by an ϭ n 29 Write the first six terms of the sequence whose nth term is (Ϫ1)n (2n ϩ 1) 30 Write the first five terms of the sequence whose nth term is an ϭ (Ϫ1)n (n2) 31 Write the first four terms and the 10th term of the sequence whose nth term is nϪ4 an ϭ n(n ϩ 2) 48 49 50 51 32 Write the sixth term of the sequence whose nth term is an ϭ n(n Ϫ 1) nϩ3 ARITHMETIC SEQUENCES In Problems 33–36, (a) identify d and a1 and (b) write the next three terms 33 2, 5, 8, … 34 3, 9, 15, … 36 2, 2.75, 3.5, 35 3, 92 , 6, 37 Find the 83rd term of the arithmetic sequence with first term and common difference Ϫ12 38 Find the 66th term of the arithmetic sequence with first term 12 and common difference Ϫ13 39 Find the 100th term of the arithmetic sequence with first term and eighth term 19 40 Find the 73rd term of the arithmetic sequence with first term 20 and 10th term 47 41 Find the sum of the first 38 terms of the arithmetic sequence with first term and 38th term 113 42 Find the sum of the first 56 terms of the arithmetic sequence with first term and 56th term 226 52 53 Observe the pattern and write three more terms of the sequence Salaries Suppose you are offered a job with a relatively low starting salary but with a $3000 raise for each of the next years How much more than your starting salary would you be making in the eighth year? Profit A new firm loses $4000 in its first month, but its profit increases by $800 in each succeeding month for the next year What is its profit in the 12th month? Pay raises If you make $36,000 and get $2400 raises each year, in how many years will your salary double? Salaries Suppose you are offered two identical jobs: one paying a starting salary of $40,000 with yearly raises of $2000 and one paying a starting salary of $36,000 with yearly raises of $2400 Which job will pay you more for your 10th year on the job? Profit A new firm loses $4000 in its first month, but its profit increases by $800 in each succeeding month for the next year What is its profit for the year? Pay raises If you are an employee, would you rather be given a raise of $2000 at the end of each year (plan I) or a raise of $600 at the end of each 6-month period (plan II)? Consider the table for an employee whose base salary is $40,000 per year (or $20,000 per 6-month period), and answer parts (a)–(g) Salary Received per 6-Month Period Period (months) Plan I Plan II 0–6 6–12 12–18 18–24 24–30 30–36 $20,000 20,000 21,000 21,000 22,000 22,000 $20,000 20,600 21,200 21,800 22,400 23,000 Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it SECTION 6.2 (a) Find the sum of the raises for plan I for the first 3 years (b) Find the sum of the raises for plan II for the first 3 years (c) Which plan is better, and by how much? (d) Find the sum of the raises in plan I for years OBJECTIVES • • • • Compound Interest; Geometric Sequences 369 (e) Find the sum of the raises in plan II for years (f) Which plan is better, and by how much? (g) Do you want plan I or plan II? 54 Pay raises As an employee, would you prefer being given a $2400 raise each year for years or a $400 raise each quarter for years? 6.2 To find the future value of and interest earned when interest is compounded at regular intervals or continuously To find the annual percentage yield (APY), or the effective annual interest rate, of money invested at compound interest To find the time it takes for an investment to reach a specified amount To find specified terms, and sums of specified numbers of terms, of geometric sequences | APPLICATION PREVIEW | A common concern for young parents is how they will pay for their children’s college educations To address this, there are an increasing number of college savings plans, such as the so-called “529 plans,” which are tax-deferred Suppose Jim and Eden established such a plan for their baby daughter Maura If this account earns 9.8% compounded quarterly and if their goal is to have $200,000 by Maura’s 18th birthday, what would be the impact of having $10,000 in the account by Maura’s first birthday? (See Example 3.) A compound interest investment (such as Jim and Eden’s account) is one in which interest is paid into the account at regular intervals In this section we consider investments of this type and develop formulas that enable us to determine the impact of making a $10,000 investment in Maura’s college tuition account by her first birthday In the previous section we discussed simple interest A second method of paying interest is the compound interest method, where the interest for each period is added to the principal before interest is calculated for the next period With compound interest, both the interest added and the principal earn interest for the next period With this method, the principal grows as the interest is added to it Think of it like building a snowman You start with a snowball the size of your fist (your investment), and as you roll the snowball around the yard, the snowball picks up additional snow (interest) The more you roll it, the snowball itself (principal) keeps growing, and as it does, the amount of snow it picks up along the way (interest) also grows This method is used in investments such as savings accounts and some U.S government bonds © Pixel Images/Shutterstock.com Compound Interest Compound Interest; Geometric Sequences Copyright 2016 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it ... of each 6-month period (plan II)? Consider the table for an employee whose base salary is $40,000 per year (or $20,000 per 6-month period), and answer parts (a)–(g) Salary Received per 6-Month... address this, there are an increasing number of college savings plans, such as the so-called “529 plans,” which are tax-deferred Suppose Jim and Eden established such a plan for their baby daughter...368 CHAPTER Mathematics of Finance 25 Bill Casler bought a $2000, 9-month certificate of deposit (CD) that would earn 8% annual simple interest Three months before