Download 7-4 Exponential Models in Recursive Form

7-4
Exponential Models in Recursive Form
TEKS FOCUS
VOCABULARY
TEKS (5)(B) Formulate exponential and
logarithmic equations that model realworld situations, including exponential
relationships written in recursive notation.
ĚExplicit formula – An explicit
TEKS (1)(D) Communicate mathematical
ideas, reasoning, and their implications
using multiple representations, including
symbols, diagrams, graphs, and language as
appropriate.
ĚRecursive formula – A recursive
formula describes the nth
term of a sequence using the
number n.
formula relates each term of a
sequence after the first term to
the term before it.
ĚSequence – A sequence is an
ordered list of numbers.
Additional TEKS (1)(A)
ĚTerm of a sequence – Each
number in a sequence is a term
of the sequence.
ĚImplication – a conclusion that
follows from previously stated
ideas or reasoning without
being explicitly stated
ĚRepresentation – a way to
display or describe information.
You can use a representation to
present mathematical ideas and
data.
ESSENTIAL UNDERSTANDING
If the numbers in a list follow a pattern, you may be able to use a rule to relate each
number in the list to its numerical position.
Key Concept
Exponentiation in Recursive Form
Sometimes you can see the pattern in a sequence by comparing each term to the one
that came before it. For example, in the sequence 133, 130, 127, 124, . . . , each term
after the first term is equal to three less than the previous term.
A recursive definition for this sequence contains two parts.
(a) an initial condition (the value of the first term): a1 = 133
(b) a recursive formula (relates each term after the first term to the one before it):
an = an-1 - 3, for n 7 1
You can use properties of exponents to write patterns based on exponentiation in
recursive form.
a1 = x 1 = x
an = xn = x x(n-1) = x an-1
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Lesson 7-4
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Exponential Models in Recursive Form
Problem 1
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TEKS Process Standard (1)(D)
Formulating Recursive Exponential Functions
The table shows the number of subscribers to an artist’s photo-sharing account
over a period of several months. Write an explicit formula and a recursive formula
to model the data.
Month
Subscribers
1
2
3
4
5
7
21
63
189
567
Step 1 Write an explicit formula.
Write each term of the sequence with an exponent and look for a pattern.
a1 = 7
=7
How can you check
the formula?
Check that the formula
works for specific values
of n, including n = 1. For
n = 1, a1 = 7(3)1-1 =
7(3)0 = 7.
#
a2 = 21
30
=7
#
a3 = 63
31
=7
#
a4 = 189
32
=7
#
a5 = 567
33
=7
# 34
In each term, the exponent is 1 less than the term number.
The explicit formula is an = 7(3)n-1 .
Step 2 Write a recursive formula.
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Write the first few terms of the sequence using the previous term and look
for a pattern.
a1 = 7
# 7 = 3a1
a3 = 63 = 3 # 21 = 3a2
a4 = 189 = 3 # 63 = 3a3
a2 = 21 = 3
The recursive formula is a1 = 7 and an = 3an-1 .
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287
Problem 2
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Writing Exponential Functions in Recursive Form
The function y = 8 ~ 2x models the number of E. coli cells y in a petri dish x hours
after the start of an experiment. Write a recursive formula to model the situation.
Let an represent the number of cells after n hours. Write the first few terms of the
sequence using the previous term and look for a pattern.
What does the
recursive formula
tell you about the
situation?
The formula says that
there are 16 cells after
the first hour and the
number of cells doubles
every hour after that.
# 21 = 16
a2 = 8 # 22 = 32 = 2 # 16 = 2a1
a3 = 8 # 23 = 64 = 2 # 32 = 2a2
a4 = 8 # 24 = 128 = 2 # 64 = 2a3
a1 = 8
The recursive formula is a1 = 16 and an = 2an-1 .
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Problem
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bl
3
TEKS Process Standard (1)(A)
Using a Recursive Exponential Function to Model a Situation
You invest $500 in a savings account that pays 2% annual interest. Write an
explicit formula and a recursive formula to model the situation.
How do you find
the amount in the
account after 1 year?
After 1 year, you have the
starting amount plus 2%
of the starting amount,
or 500 + 500(0.02) =
500(1.02).
Step 1 Define the variables and write the first few terms of the sequence.
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Let n represent the number of years. Let an represent the amount in the
account after n years.
# 1.02
a2 = 500 # 1.02 # 1.02
a3 = 500 # 1.02 # 1.02 # 1.02
a1 = 500
Step 2 Write an explicit formula.
In the above expressions for an , the factor 1.02 appears n times.
an = 500(1.02)n
Step 3 Write a recursive formula.
In the above expressions for an , a1 = 500
1.02 times the previous term.
a1 = 510 and an = 1.02an-1
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Lesson 7-4
Exponential Models in Recursive Form
# 1.02 = 510 and each term is
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PRACTICE and APPLICATION EXERCISES
For additional support when
completing your homework,
go to PearsonTEXAS.com.
Scan page for a Virtual Nerd™ tutorial video.
1. The table shows the number of visitors to a
Web page over a period of several months.
Write an explicit formula and a recursive
formula to model the data.
Month
Visitors
2. The table shows the profit made by a small
catering company over a period of several
years. Write an explicit formula and a
recursive formula to model the data.
1
2
3
4
5
17
34
68
136
272
2
3
4
5
1
Year
Profit ($) 5200
3. A student repeatedly folds a sheet of paper
in half. The table shows the area of each of
the congruent regions formed by the creases.
Write an explicit formula and a recursive
formula to model the data.
Number of Folds 1
Area of Each
256
Region (cm2)
7800 11,700 17,550 26,325
1 fold
2 congruent
regions
2
3
4
5
128
64
32
16
2 folds
4 congruent
regions
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4. The function y = 25 2x models the jackpot y, in dollars, on a game show
after the show has been on the air for x weeks. Write a recursive formula to
model the situation.
5. The function y = 3500(1.1)x models the value y, in dollars, of a piece of artwork
after x years. Write a recursive formula to model the situation.
6. The function y = 18,000(0.85)x models the value y, in dollars, of a car after x years.
Write a recursive formula to model the situation.
7. You invest $400 in a savings account that pays 2.5% annual interest. Write an
explicit formula and a recursive formula to model the situation.
8. An employee joins a company at the start of the year and earns a salary of $40,000.
At the end of each year, the employee receives a 4% raise. Write an explicit formula
and a recursive formula to model the situation.
9. An accountant buys a new computer for $1200. Each year, the value of the
computer decreases by 20%. Write an explicit formula and a recursive formula to
model the situation.
10. Explain Mathematical Ideas (1)(G) The value of a collectible baseball card
is currently $620 and its value is expected to increase by 5% each year. A
student modeled the situation by writing the recursive formula a1 = 620 and
an = (1.05)an-1 , where an represents the value of the baseball card after n years.
Is the student’s formula correct? Explain.
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11. The fractal known as the Sierpinski carpet begins
with a square. At each subsequent stage, every
square is divided into nine congruent squares and
the center square is removed. Assume the area of the
square in Stage 1 is 1 square unit. Write an explicit
formula and a recursive formula to model the area
an of the figure in the nth stage of the fractal.
Stage 1
Stage 2
12. The owner of a corner store finds that a
1
Week
juice drink suddenly becomes popular with
Revenue From
students at a school across the street. The
4
Juice Drinks ($)
store’s owner records the revenue from the
drinks over a period of several weeks, but
she does not do so every week. The table shows the data.
Write an explicit formula and a recursive formula to model the data.
3
4
6
36
108
972
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13. Analyze Mathematical Relationships (1)(F) A real-world situation is modeled
by the explicit formula an = p qn-1 for real numbers p and q with p ≠ 0, q ≠ 0,
and q ≠ 1. Write a recursive formula to model the situation.
TEXAS Test Practice
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14. The amount of money in a bank account is modeled by the explicit formula
an = 320(1.035)n-1 , where an is the amount of money in the account, in dollars,
at the end of n years. Which is a true statement about the situation?
A. The initial investment in the account is $331.20.
B. The account pays 35% annual interest.
C. In a recursive formula for the situation, a1 = 331.2.
D. The amount of money in the account at the end of year 5 is $367.21.
15. Which of the following is a recursive formula for an exponential relationship?
F. a1 = 2 and an = 4 + an-1
G. a1 = 4 and an = 0.5an-1
H. a1 = 0.5 and an = n an-1
J. a1 = 4 and an = (an-1)2
16. The number of employees at a small software company grows according to
an exponential relationship. After 2 years, the company has 18 employees.
After 4 years, the company has 72 employees. If an represents the number of
employees after n years, which recursive formula models the situation?
A. a1 = 18 and an = 4an-1
C. a1 = 9 and an = 2an-1
B. a1 = 18 and an = 2an-1
D. a1 = 9 and an = 4an-1
17. Describe a real-world situation that can be modeled by the recursive formula
a1 = 6 and an = 3an-1 .
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Lesson 7-4
Exponential Models in Recursive Form
Stage 3