6.7 geometric sequences - big ideas math · geometric sequences in this lesson, you will ... math...

306 Chapter 6 Exponential Equations and Functions

Geometric Sequences6.7

How are geometric sequences used to

describe patterns?

Work with a partner.

● Enter the keystrokes on a calculator and record the results in the table.

● Describe the pattern.

a. Step 1 2

Step 2 2

Step 3 2

Step 4 2

Step 5 2

Step 1 2 3 4 5

Calculator Display

b. Step 1 46

Step 2 5.

Step 3 5.

Step 4 5.

Step 5 5.

c. Use a calculator to make your own sequence. Start with any number and multiply by 3 each time. Record your results in the table.

Step 1 2 3 4 5

Calculator Display

ACTIVITY: Describing Calculator Patterns11

ME MM

CE78

945

612

30.

OFFON/C

%

Step 1 2 3 4 5

Calculator Display

COMMON CORE

Geometric Sequences In this lesson, you will● extend and graph

geometric sequences.● write equations for

geometric sequences.● solve real-life problems.Learning StandardsF.BF.2F.IF.3F.LE.2

Section 6.7 Geometric Sequences 307

4. IN YOUR OWN WORDS How are geometric sequences used to describe patterns? Give an example from real life.

Use what you learned about geometric sequences to complete Exercise 4 on page 310.

Work with a partner. A sheet of paper is about 0.1 mm thick.

a. How thick would it be if you folded it in half once?

b. How thick would it be if you folded it in half a second time?

c. How thick would it be if you folded it in half 6 times?

d. What is the greatest number of times you can fold a sheet of paper in half? How thick is the result?

e. Do you agree with the statement below? Explain your reasoning.

“If it were possible to fold the paper 15 times, it would be taller than you.”

ACTIVITY: Folding a Sheet of Paper22

The King and the Beggar

A king offered a beggar fabulous meals for one week. Instead, the beggar asked for a single grain of rice the fi rst day, 2 grains the second day, and double the amount each day after for one month. The king agreed. But, as the month progressed, he realized that he would lose his entire kingdom.

Work with a partner.

● Why does the king think he will lose his entire kingdom?

● Write your own story about doubling or tripling a small object many times.

● Draw pictures for your story.

● Include a table to organize the amounts.

● Write your story so that one of the characters is surprised by the size of the fi nal number.

ACTIVITY: Writing a Story33

half 6 times?

f ld h t f i h

Repeat CalculationsWhat calculations are repeated? How does this help you answer the question?

Math Practice


Lesson6.7Lesson Tutorials

Key Vocabularygeometric sequence, p. 308common ratio, p. 308

Geometric Sequence

In a geometric sequence, the ratio between consecutive terms is the same. This ratio is called the common ratio. Each term is found by multiplying the previous term by the common ratio.

1, 5, 25, 125, . . . Terms of a geometric sequence

× 5 × 5 × 5 Common ratio

Exercises 11–16

EXAMPLE Extending a Geometric Sequence11

Write the next three terms of the geometric sequence 3, 6, 12, 24, . . . .

Use a table to organize the terms and extend the pattern.

Position 1 2 3 4 5 6 7

Term 3 6 12 24 48 96 192

× 2 × 2 × 2 × 2 × 2 × 2

The next three terms are 48, 96, and 192.

Each term is twice the previous term. So, the common ratio is 2.

EXAMPLE Graphing a Geometric Sequence22

Graph the geometric sequence 32, 16, 8, 4, 2, . . . . What do you notice?

Make a table. Then plot the ordered pairs (n, an).

Position, n 1 2 3 4 5

Term, an32 16 8 4 2

The points of the graph appear to lie on an exponential curve.

Write the next three terms of the geometric sequence. Then graph the sequence.

1. 1, 3, 9, 27, . . . 2. 64, 16, 4, 1, . . . 3. 80, − 40, 20, − 10, . . .

n

an

36

32

28

24

20

16

12

8

4

00 1 2 3 4 5

(1, 32)

(2, 16)

(3, 8)(4, 4)

(5, 2)

Multiply a term by 2 to fi nd the next term.


EXAMPLE Real-Life Application33

Clicking the zoom-out button on a mapping website doubles the side length of the square map.

a. Write an equation for the nth term of the geometric sequence.

The fi rst term is 5 and the common ratio is 2.

an = a1r n − 1 Equation for a geometric sequence

an = 5(2)n − 1 Substitute 5 for a1 and 2 for r.

b. Find and interpret a8.

Use the equation to fi nd the 8th term.

an = 5(2)n − 1 Write the equation.

= 5(2)8 − 1 Substitute 8 for n.

= 640 Simplify.

The side length of the square map after 8 clicks is 640 miles.

4. WHAT IF? After how many clicks on the zoom-out button is the side length of the map 2560 miles?Exercises 25–28

Because consecutive terms of a geometric sequence change by equal factors, the points of any geometric sequence with a positive common ratio lie on an exponential curve. You can use the fi rst term and the common ratio to write an exponential function that describes a geometric sequence.

Position, n Term, an Written using a1 and r Numbers

1 fi rst term, a1 a1 1 2 second term, a2 a1r 1 ⋅ 5 = 5 3 third term, a3 a1r 2 1 ⋅ 52 = 25 4 fourth term, a4 a1r 3 1 ⋅ 53 = 125

… … … …

n nth term, an a1r n − 1 1 ⋅ 5n − 1

Equation for a Geometric Sequence

Let an be the nth term of a geometric sequence with fi rst term a1 and common ratio r. The nth term is given by

an = a1r n − 1.

Study TipNotice that an = a1r n − 1 is of the

form y = abx.

Zoom-out Clicks 1 2 3

Map Side Length (miles)

5 10 20

Exercises6.7


9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=

1. WRITING How are arithmetic sequences and geometric sequences different?

2. REASONING Compare and contrast the two sequences.

2, 4, 6, 8, 10, . . . 2, 4, 8, 16, 32, . . .

3. CRITICAL THINKING Why do the points of a geometric sequence lie on an exponential curve only when the common ratio is positive?

4. Enter 4 on a calculator. Multiply by 6 four times. Record your results in the table. Describe the pattern.

Step 1 2 3 4 5

Calculator Display

Find the common ratio of the geometric sequence.

5. 3, − 12, 48, − 192, . . . 6. 200, 100, 50, 25, . . . 7. 7640, 764, 76.4, 7.64, . . .

8. 9, − 18, 36, − 72, . . . 9. 0.1, 0.9, 8.1, 72.9, . . . 10. 5, 1, 1

— 5

, 1

— 25

, . . .

Write the next three terms of the geometric sequence. Then graph the sequence.

11. 2, 10, 50, 250, . . . 12. − 7, 14, − 28, 56, . . . 13. 81, − 27, 9, − 3, . . .

14. − 375, − 75, − 15, − 3, . . . 15. 36, 6, 1, 1

— 6

, . . . 16. 1

— 49

, 1

— 7

, 1, 7, . . .

17. ERROR ANALYSIS Describe and correct the error in writing the next three terms of the geometric sequence.

18. BADMINTON A badminton tournament begins with 128 teams. After the fi rst round, 64 teams remain. After the second round, 32 teams remain. How many teams remain after the third, fourth, and fi fth rounds?

Tell whether the sequence is geometric, arithmetic, or neither.

19. − 8, 0, 8, 16, . . . 20. − 1, 3, − 5, 7, . . . 21. 1, 4, 9, 16, . . .

22. 3

— 49

, 3

— 7

, 3, 21, . . . 23. 192, 24, 3, 3

— 8

, . . . 24. − 25, − 18, − 12, − 7, . . .

22

−8, 4, −2, 1, . . .

×(−2) ×(−2) ×(−2)

The next three terms are −2, 4, and −8.

✗

11

the ge

Tell wheth

Help with Homework


Simplify the expression. (Skills Review Handbook)

34. 2n − 6n 35. 2(4x − 5) + x 36 4(y − 1) − (y + 2)

37. MULTIPLE CHOICE What is the solution of 6(3 − x) = − 4x + 12? (Section 1.3)

○A x = − 3 ○B x = − 2 ○C x = 3 ○D x = 6

Write an equation for the nth term of the geometric sequence. Then fi nd a7.

25. 1, − 5, 25, − 125, . . . 26. 2, 8, 32, 128, . . .

27. n 1 2 3 4

an 5 15 45 135

28. n 1 2 3 4

an 2 14 98 686

29. CHAIN EMAIL You start a chain email and send it to 6 friends. The process continues and each of your friends forwards the email to 6 people.


b. Describe the domain. Is the domain discrete or continuous?

30. REASONING What is the 9th term of a geometric sequence where a3 = 81 and r = 3?

31. PRECISION Are the terms of a geometric sequence independent or dependent? Explain your reasoning.

32. ROOM AND BOARD A college student makes a deal with her parents to live at home instead of living on campus. She will pay her parents $0.01 for the fi rst day of the month, $0.02 for the second day, $0.04 for the third day, and so on.


b. What will she pay on the 25th day?

c. Did the student make a good choice or should she have chosen to live on campus? Explain.

33. RepeatedReasoningRepeatedReasoning A soup kitchen makes 16 gallons of soup.

Each day, a quarter of the soup is served and the rest is saved for the next day.

a. Write the fi rst fi ve terms of the sequence of the number of fl uid ounces of soup left each day.

b. Write an equation to represent the sequence.

c. When is all the soup gone? Explain.

33

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