subtraction) generator. The number added to each term to get the next term is called the common difference.
A geometric sequence is a sequence with a multiplication (or division) generator. The number multiplied by each term to get the next term is called the common ratio or the multiplier.
2-86. Convert each percent increase or decrease into a multiplier. [ a: 1.03, b: 0.75, c: 0.87, d: 1.0208 ]
a. 3% increase b. 25% decrease
c. 13% decrease d. 2.08% increase
2-87. Mr. C is such a mean teacher! The next time Mathias gets in trouble, Mr. C has designed a special detention for him. Mathias will have to go out into the hall and stand exactly 100 meters away from the exit door and pause for a minute. Then he is allowed to walk exactly halfway to the door and pause for another minute. Then he can again walk exactly half the remaining distance to the door and pause again, and so on. Mr. C says that when Mathias reaches the door he can leave, unless he breaks the rules and goes more than halfway, even by a tiny amount. When can Mathias leave? Prove your answer using multiple representations. [ Technically, Mathias can never leave, either because he will never reach the door or because he cannot avoid breaking the rules. The equation for this situation is y=100(0.5)x, where x is the
number of minutes that have passed and y is the distance (in meters) from the door. ]
2-88. Simplify each expression. [ a: 8m5, b: 2y3, c: ! 2
3y5 , d: !8x 6 ] a. (2m3)(4m2) b. 6y5 3y2 c. !4y2 6y7 d. (–2x 2)3
2-89. Without a calculator, perform each operation below. [ a: 11 12 , b: 8+3x 12 , c: 2x3+x3, d: 2x+3y xy ] a. 23+ 14 b. 23+ 4x c. 23+ 1x d. 2y+ 3x
2-90. Factor each expression below. [ a: 3y(y+2), b: (w!2)(w!3), c: (x+2)(x!2), d: (3x+2)(3x!2) ]
a. 3y2+6y b. w2!5w+6
c. x2!4 d. 9x2!4
2-91. Solve the system of equations at right. [ (2, –4) ] y=!x!2
Lesson 2.1.7 Is it a function?
Comparing Sequences and Functions
Lesson Objective: Students will recognize that sequences are functions with domains
limited to non-negative integers. Students will use Guess and Check or graphical methods to solve exponential equations.
Length of Activity: One day (approximately 50 minutes)
Core Problems: Problems 2-92 through 2-94
Ways of Thinking: Justifying, generalizing, choosing a strategy
Materials: None
Suggested Lesson Activity:
Have teams read the lesson introduction and the target questions for the lesson and then start on problems 2-92 through 2-94. While working on problem 2-92, teams should recognize that the number 400 is not a term in the given sequence, but it is a possible output of the corresponding linear function. As students compare and contrast the sequence and function in multiple representations, they should notice that the sequence is discrete, while the corresponding function is continuous. In problem 2-93, teams should again recognize that the number 1400 is not a term in the sequence. In part (b), they will need to solve the equation 1400=2!3x. Encourage discussion about possible ways to solve this. If teams are stuck, you might ask, “Could x = 10? What about 5? What numbers does x have to be between? How do you know?” Students may decide to solve it graphically or they may use Guess and Check. If more than one strategy comes up, be sure to allow time for teams to share their strategies with the class.
After teams have had some time to discuss problem 2-94, bring them together and lead a whole-class discussion. It is not uncommon for students to claim that a sequence is not a function because it is not continuous. This is an opportunity to revisit the idea of what a function is. Help students recognize that a sequence is a function; it just has a restricted domain. Then give them a few minutes to complete a
Learning Log entry, in which they will answer the questions posed at the beginning of the lesson.
If time permits, direct teams to continue on problems 2-95 through 2-97.
Closure: (10 minutes)
The Learning Log entry in problem 2-94 provides good closure to the
ideas for the day. Ask a few volunteers to share their ideas with the
class. If teams have finished problem 2-97, give them a chance to share
x y
2.1.7 Is it a function?
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
Comparing Sequences and Functions
Student pages for this lesson are 85 – 88.
Throughout this chapter, you have been learning about sequences. In Chapter 1, you started to learn about functions. But what is the difference? In this lesson, you will compare and contrast sequences with functions. By the end of the lesson, you will be able to answer these questions:
Is a sequence different from a function?
What is the difference between a sequence t(n) and the function f(x) with the same rule?
2-92. Consider sequence t(n) below. [ a: See tableand graph below right, t(n)=4n!5; b: no, because the domain of t(n) includes only nonnegative integers, none of which gives an output value of 400; c: The graph of f(x) is continuous and t(n) is discrete (see table and graph far below right); d: Yes, f(x) can be equal to 400 (when x= 101.25),because it has a domain of all real numbers. ]
–5, –1, 3, 7, … a. Create multiple representations of the
sequence t(n).
b. Is it possible for the equation representing t(n) to equal 400? Justify your answer. c. Create multiple representations of the
function f(x)=4x!5. How are f(x) and
t(n) different? How can you show their differences in each of the representations?
d. For the function f(x)=4x!5, is it possible for f(x) to equal 400? Explain.
x –4 –3 –2 –1 0 1 2 3 4 f(x) –21 –17 –13 –9 –5 –1 3 7 11 n 0 1 2 3 4 t(n) –5 –1 3 7 11 n t(n)
2-93. Let us consider the difference between t(n)=2!3n and f(x)=2!3x. [ a: No, it is not possible because t(5)=486and t(6)=1458 and n is restricted to non-negative integer values; b: Yes, it is possible because f(x)=1400 when x!5.963. c: The functions use the same equation, but t(n) is discrete and f(x) is continuous. ]
a. Is it possible for t(n) to equal 1400? If so, find the value of n that makes t(n)=1400. If not, justify why not.
b. Is it possible for f(x) to equal 1400? If so, find the value of x that
makes f(x)=1400. Be prepared to share your solving strategy with the class. c. How are the two functions similar? How are they different?
2-94. LEARNING LOG
Is a sequence a function? Justify your answer completely. If so, what makes it different from the functions that are usually written in the form f(x)=_________? If not, why not? Be prepared to share your ideas with the class. After a class discussion about these questions, answer the questions in your Learning Log. Title this entry
“Sequences vs. Functions” and label it with today’s date. [ Yes, a sequence is a function, because each input has only one outputut. However, its domain is limited to non-negative integer values of x. ]
2-95. Janine was working on her homework but lost part of it. She knew that one output of p(r)=2!5r is 78,000, but she could not remember if p(r) is a sequence or if it’s a regular function. With your team, help her figure it out. Be sure to justify your decision. [ p(6)=31250 and p(7)=156250; It must be a function because no term is equal to 78,000. ]
2-96. Solve each of the following equations for x, accurate to the nearest 0.01. [ a: x=6, b: x!3.61 ]
a. 200(0.5)x =3.125 b. 318=6!3x
2-97. Khalil is working with a geometric sequence. He knows that t(0)= 3 and that the sum of the first three terms (t(0), t(1), and t(2)) is 63. Help him figure out the sequence. Be prepared to share your strategies with the class. [ The most likely answer is
An exponential function has the general form y=kmx, where k is the initial value and m>0 is the multiplier. The graph of an exponential function is continuous. Be careful: The independent variable
x has to be in the exponent. For example, y=x2 is not an exponential equation, even though it has an exponent.
The number by which you multiply a quantity to increase or decrease it by a given percentage is called the multiplier for that percentage. For
example, the multiplier for an increase of 7% is 1.07. The multiplier for a decrease of 7% is 0.93.
2-98. Is it possible for the sequence t(n)=5!2n to have a term with the value of 200? If so, which term is it? If not, justify why not. [ No; the 5th
term is 160, and the 6th
term is 320. Justifications vary. ]
2-99. Is it possible for the function f(x)=5!2x to have an output of 200? If so, what input gives this output? If not, justify why not. [ yes, x!5.322 ]
2-100. Consider the following sequences as you complete parts (a) through (c) below.
[ a: Sequence 1: 10, 14, 18, 22, add 4, t(n)=4n+2; Sequence 2: 0, !12,
!24,!36, subtract 12, t(n) =!12n+24; Sequence 3: 9, 13, 17, 21, add
4, t(n)=4n+1; b: yes, Sequence 1: 18, 54, 162, 486, multiply by 3; Sequence 2: 6, 3, 1.5, 0.75, multiply by1
2 ; Sequence 3: 25, 125, 625, 3125, multiply by 5;
c: Answers vary, but the point is to have students create their own rule and write terms that correspond to it. ]
Sequence 1 Sequence 2 Sequence 3
2, 6, … 24, 12, … 1, 5, …