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 sixtyfourth 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
s_{1} = 16t^{2} + 16, s_{1} = 0 if t = 1
s_{2} = 16t^{2} + 16(0.81), s_{2} = 0 if t = 0.9
s_{3} = 16t^{2} + 16(0.81)^{2}, s_{3} = 0 if t = 0.9^{2}
s_{4} = 16t^{2} + 16(0.81)^{3}, s_{4} = 0 if t = 0.9^{3}
.
.
.
s_{n} = 16t^{2} + 16(0.81)^{n1}, s_{n} = 0 if t = 0.9^{n1}
beginning with s_{2} 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.
