306 Chapter 6 Exponential Equations and Functions Geometric Sequences 6.7 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 4 6 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 Patterns 1 1 ME M M CE 7 8 9 4 5 6 1 2 3 0 . OFF ON/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 Standards F.BF.2 F.IF.3 F.LE.2
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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
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
308 Chapter 6 Exponential Equations and Functions
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.
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
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.
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.
a. Write an equation for the nth term of the geometric sequence.
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.
a. Write an equation for the nth term of the geometric sequence.
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.