GEOMETRY The sides of a square are 27 inches in length.

A sequence is called a ________ sequence if the ratios between consecutive terms are the same. This ratio is called the ________ ratio.

The th term of a geometric sequence has the form ________.

The formula for the sum of a finite geometric sequence is given by ________.

The sum of the terms of an infinite geometric sequence is called a ________ ________.

2,

3, 12, 21, 30, . . .

25, 20, 15, 10, . . .

2, 1 4, 1 8, . . .

5, 1, 0.2, 0.04, . . .

8, 1 4, 1 2, 1, . . .

9, 6, 4, 8 3, . . . 1

1, , . . . 1 2, 1 3, 1 4, . . .

1 5, 2 7, 3 9, 4 11 1, , . . .

1, 7, 7, 77, . . . , .

2, 4 3 , 8 3 , 16 33 1,

a1  4, r  3

a1  8, r  2

a1  1, 3 r  1 2

a1  1, r  1 a1  1, 3

a1  5, 4 r  1 10

a1  6, r  1 a1  5, 4 r

a1  1, r  e

a1  2, r

a1  3, r  5

a1  4, r  1 2 a

a a1  5, r  2x 1  2, r  x 4

a1  5, r  2x

a1  64, ak 1  3ak 1 2ak

a1  81, ak 1  1 a1  64, ak 1  3ak

a1  9, ak 1  2ak

a1  5, ak 1  2ak

a1  6, ak 1  2 ak 3 2ak

a1  80, ak 1  1 a1  6, ak 1  2 ak 3

a1  4, r  , n  8 1 2, n  10

a1  5, r  7 2 a1  4, r  , n  8 1

a1  6, r  , n  10 1 3, n  12

a1  64, r  1 4 a1  6, r  , n  10 1

a1  100, r  e , n  4 x , n  9

a1  1, r  e

a1  1, r  2, n  12 a

a1  1, r  3, n  8 a

a1  500, r  1.02, n  40

a1  1000, r  1.005, n  60

9th term: 11, 33, 99, . . .

7th term: 3, 36, 432, . . .

10th term: 5, 30, 180, . . .

22nd term: 4, 8, 16, . . .

8th term:

7th term:

3rd term:

1st term:

6th term:

7th term:

5th term:

9th term

an  18 2 3 n1

an  18 2 3 n1 an

an  18 3 2 n1 a

an  18 3 2 n1 a

an  12 0.75n1 Ye

an  10 1.5n1 an

an  12 0.4n1 an

an  20 1.25n1 an

an  2 1.3n1 a

an  10 1.2n1 an

7 n1 4n1 a

10 n1 3 2 n1

6 n1 7n1

8 n1 5 5 2 n1

7 i1 64 1 2 i1

10 i1 2 1 4 i1

6 i1 32 1 4 i1

12 i1 16 1 2 i1

20 n0 3 3 2

40 n0 5 3 5 n

15 n0 2 4 3 n

20 n0 10 1 5 n

5 n0 300 1.06n

6 n0 500 1.04n

40 n0 2 1 4 n

50 n0 10 2 3 n1

10 i1 8 1 4 i1

25 i0 8 1 2 i

10 i1 5 1 3 i1

100 i1 15 2 3 i1

10 30 90 ... 7290

9 18 36 ... 1152

2 1 2 1 8 ... 1 2048 9

15 3 3 5 ... 3 625 2

0.1 0.4 1.6 ... 102.4

32 24 18 ... 10.125

n0 1 2

n0 2 2 3

n0 1 2

n0 2 2 3

n0 4 1 4

n0 1 10

n0 0.4n

n0 4 0.2n

n0 3 0.9n

n0 10 0.2n

8 6 3 ... 9 2 27 8 . . .

9 6 2004

9 1 3 1 3 . . . 9

125 36 25 6 5 6 . . . 1

0.36

0.297

0.318 1

1.38

f x  6 1 0.5x 1 0.5 , 0

f x  2 1 0.8x 1 0.8 ,

DATA ANALYSIS: POPULATION The table shows the mid-year populations of China (in millions) from 2002 through 2008. (Source: U.S. Census Bureau) (a) Use the exponential regression feature of a graphing utility to find a geometric sequence that models the data. Let represent the year, with corresponding to 2002. (b) Use the sequence from part (a) to describe the rate at which the population of China is growing. (c) Use the sequence from part (a) to predict the population of China in 2015. The U.S. Census Bureau predicts the population of China will be 1393.4 million in 2015. How does this value compare with your prediction? (d) Use the sequence from part (a) to determine when the population of China will reach 1.35 billion.

COMPOUND INTEREST A principal of $5000 is invested at 6% interest. Find the amount after 10 years if the interest is compounded (a) annually, (b) semiannually, (c) quarterly, (d) monthly, and (e) daily

COMPOUND INTEREST A principal of $2500 is invested at 2% interest. Find the amount after 20 years if the interest is compounded (a) annually, (b) semiannually, (c) quarterly, (d) monthly, and (e) daily.

DEPRECIATION A tool and die company buys a machine for $175,000 and it depreciates at a rate of 30% per year. (In other words, at the end of each year the depreciated value is 70% of what it was at the beginning of the year.) Find the depreciated value of the machine after 5 full years.

ANNUITIES A deposit of $100 is made at the beginning of each month in an account that pays 6% interest, compounded monthly. The balance in the account at the end of 5 years is

ANNUITIES A deposit of $50 is made at the beginning of each month in an account that pays 8% interest, compounded monthly. The balance in the account at the end of 5 years is

ANNUITIES A deposit of dollars is made at the beginning of each month in an account with an annual interest rate compounded monthly. The balance after years is

ANNUITIES A deposit of dollars is made at the beginning of each month in an account with an annual interest rate compounded continuously. The balance after years is

P  $50, r  5%, t  20 years

P  $75, r  3%, t  25 years

P  $100, r  2%, t  40 years

P  $20, r  4.5%, t  50 years

ANNUITIES Consider an initial deposit of dollars in an account with an annual interest rate compounded monthly. At the end of each month, a withdrawal of dollars will occur and the account will be depleted in years. The amount of the initial deposit required is Show that the initial deposit is

ANNUITIES Determine the amount required in a retirement account for an individual who retires at age 65 and wants an income of $2000 from the account each month for 20 years. Use the result of Exercise 125 and assume that the account earns 9% compounded monthly

$400 75%

$250 80%

$600 72.5%

$450 77.5%

GEOMETRY The sides of a square are 16 inches in length. A new square is formed by connecting the midpoints of the sides of the original square, and two of the resulting triangles are shaded (see figure). If this process is repeated five more times, determine the total area of the shaded region.

GEOMETRY The sides of a square are 27 inches in length. New squares are formed by dividing the original square into nine squares. The center square is then shaded (see figure). If this process is repeated three more times, determine the total area of the shaded region.

SALARY An investment firm has a job opening with a salary of $45,000 for the first year. Suppose that during the next 39 years, there is a 5% raise each year. Find the total compensation over the 40-year period.

SALARY A technology services company has a job opening with a salary of $52,700 for the first year. Suppose that during the next 24 years, there is a 3% raise each year. Find the total compensation over the 25-year period.

DISTANCE A bungee jumper is jumping off the New River Gorge Bridge in West Virginia, which has a height of 876 feet. The cord stretches 850 feet and the jumper rebounds 75% of the distance fallen. (a) After jumping and rebounding 10 times, how far has the jumper traveled downward? How far has the jumper traveled upward? What is the total distance traveled downward and upward? (b) Approximate the total distance, both downward and upward, that the jumper travels before coming to rest.

DISTANCE A ball is dropped from a height of 16 feet. Each time it drops feet, it rebounds feet. (a) Find the total vertical distance traveled by the ball. (b) The ball takes the following times (in seconds) for each fall. Beginning with the ball takes the same amount of time to bounce up as it does to fall, and so the total time elapsed before it comes to rest is Find this total time.

A sequence is geometric if the ratios of consecutive differences of consecutive terms are the same.

You can find the th term of a geometric sequence by multiplying its common ratio by the first term of the sequence raised to the th power.

GRAPHICAL REASONING Consider the graph of

CAPSTONE

WRITING Write a brief paragraph explaining why the terms of a geometric sequence decrease in magnitude when

Find two different geometric series with sums of 4.