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### High School Mathematics - 23.14 Applications - Geometric Sequence/Series

 Geometric Series: Story Problem: A fable says that a wise women solved a king's dire problem and the king said she could have anything she wanted in return. She demurely asked only for a chessboard with 1 grain of wheat on the first square, 2 on the second, 4 on the third, 8 on the Fourth and so on up to sixty-fourth square. Find a formula for the nth term of the arithmetic sequence: -2, -8, -14, -20, ..... Calculate the missing terms in the arithmetic sequence, if a1 = 6, and a5 = 66. Salary: Kim's starting salary was \$40,000 per year. Each year we receive a cost of living adjustment (COLA) of three percent of our original salary. Write a sequence showing my salary for my first five years of working here. Solution We have a a1 = 40,000 a2 = 40,000(1.03) = 41,200 a3 = 41,000(1.03) = 40,000(1.03)2 = 42,436 a4 = 40,000(1.03)3 = 43,709.08 a5 = 40,000(1.03)4 = 45,020.35 Annuities: Determine the amount required in an individual retirement account for an individual who retires at age 65 and wants an income of \$2000 from the account each month for 20 years. Assume that the account earns 9% compounded monthly. P = W(1+r/12)-1 + W(1+r/12)-2 + ..............W(1+r/12)-12t The initial deposite is given by P = W (12/r) [1- (1+r/12)-12t Salary: A company is offering a job with a salary of \$30,000 for the first year. Suppose that during the next 39 years, there is a 5% raise each year. Determine the total compensation over the 40 year period. Bouncing ball A ball is dropped from a height of 16 feet. Each time it drops h feet, it rebounds 0.81h feet. a) Find the total distance travelled by the ball b> The ball takes the following time for each fall s1 = -16t2 + 16, s1 = 0 if t = 1 s2 = -16t2 + 16(0.81), s2 = 0 if t = 0.9 s3 = -16t2 + 16(0.81)2, s3 = 0 if t = 0.92 s4 = -16t2 + 16(0.81)3, s4 = 0 if t = 0.93 . . . sn = -16t2 + 16(0.81)n-1, sn = 0 if t = 0.9n-1 beginning with s2 the ball takes the same amount of time to bounce up as it does to fall, and thus the total time elapsed before it comes to rest is t = 1 + 2 sigma(n=1 to infinity) (0.9)n Find this total. a) 152.42 feet b) 19 seconds Directions: Answer the following questions. Also write at least 5 examples of your own.
 Q 1: Annuties: A deposite of \$100 is made at the beginning of each month in an account that pays 10% interest, compounded monthly. The balance A in the account at the end of 5 years is given by A = 100 (1+ 0.12/12)1 + ..................+ 100 (1 + 0.10/12)60. Find A. Answer: Q 2: Population growth: A city of 250,000 people is growing at a rate of 1.3% per year. Estimate the population of the city 30 years from now.Answer: Q 3: Compound Interest: A principal of \$1000 is invested at 10% interest. Find the amount after 10 years if the interest is compounded (a) anually, (b) semiannually, (c) quarterly, (d) monthly, and (e) daily. Answer: Q 4: A deposite of \$50 is made on the first day of each month in a savings account that pays 6% compounded monthly. What is the balance at the end of 2 years? \$2000.50\$1277.96\$1200.89 Question 5: This question is available to subscribers only! Question 6: This question is available to subscribers only!