Lesson 5.3.3

HW: 5-117 to 5-123

Learning Target: Scholars will recognize that all sequences are functions with domains limited to positive integers. Scholars will use graphical methods to solve exponential equations.

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 equation?

5-111. Consider sequence t(n) below.

−5, −1, 3, 7, …

  1. Createmultiple representations, namely, a table, a graph and an equation (recursive or explicit), for the sequence t(n).
  2. Is it possible for the equation representingt(n)to equal 400? Justify your answer.
  3. Create the same multiple representations as you did in part (a) for thefunction f(x) = 4x − 9. How are f(x) and t(n) different? How can you see their differences in each of the representations?
  4. For the function f(x) = 4x − 9, is it possible for f(x) to equal 400?Explain why or why not.

5-112. Let us consider the difference between t(n) = 2 ·3nand f(x) = 2 · 3x.

  1. Isf(x)= 2 · 3xa function? Why or why not?Is t(n)= 2 ·3na function?
  2. Is it possible for t(n) to equal 1400? If so, find the value of n that makes t(n) = 1400. If not, justifywhy not.
  3. 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.
  4. How are the two functions similar? How are they different?

5-114. Janine was working on her homework but lost part of it. She knew that one output of p(r) = 2 · 5ris 78,000, but she could not remember if p(r) is a sequence or if it is a regular function. With your team, help her figure it out. Be sure to justifyyour decision.

5-117. Is it possible for the sequence t(n) = 5 · 2nto have a term with the value of 200? If so, which term is it? If not, justifywhy not.

5-118. Is it possible for the function f (x) = 5 · 2xto have an output of 200? If so, what input gives this output? If not, justify why not.

5-119. Consider the following sequences as you complete parts (a) through (c) below.

Sequence 1
2, 6, … / Sequence 2
24, 12, … / Sequence 3
1, 5, …
  1. Assuming that the sequences above are arithmetic with t(1) as the first term, find the next four terms for each sequence. For each sequence, write an explanation of what you did to get the next term and write an equation for t(n).
  2. Would your terms be different if the sequences were geometric? Find the next four terms for each sequence if they are geometric. For each sequence, write an explanation of what you did to get the next termand write an equation fort(n).
  3. Create a totally different type of sequence for each pair of values shown above, based on your own equation. Write your equation clearly (using words or algebra) so that someone else will be able to find the next three terms that you want.

5-120. For the function g(x) = x3 + x2 − 6x , find the value of each expression below.

  1. g(1)
  2. g(−1)
  3. g(−2)
  4. g(10)
  5. Find at least one value of x for which g(x) = 0.
  6. If f(x) = x2 − x + 3, find g(x) − f(x).

5-121. Write equations to solve each of the following problems.

  1. When the Gleo Retro (a trendy commuter car) is brand new, it costs $23,500. Each year it loses 15% of its value. What will the car be worth when it is 15 years old?
  2. Each year the population of Algeland increases by 12%. The population is currently 14,365,112. What will the population be 20 years from now?

5-122. An arithmetic sequence has t(8) = 1056 and t(13) = 116. Write an equation for the sequence.What is t(5)?

5-123. Describe the domain of each function or sequence below. 5-123 HW eTool (Desmos).

  1. The function f(x) = 3x − 5.
  2. The sequence t(n) = 3n − 5.
  3. The function.
  4. The sequence.
  • 5-117.No; the 5th term is 160, and the 6thterm is 320. Justifications vary.
  • 5-118.Yes, x≈ 5.322.
  • 5-119. See below:
  • Sequence 1: 10, 14, 18, 22, add 4, t(n) = 4n − 2; Sequence 2: 0, −12,−24,−36, subtract 12, t(n) = −12n + 36; Sequence 3: 9, 13, 17, 21, add 4, t(n) = 4n − 3
  • Yes, Sequence 1: 18, 54, 162, 486, multiply by 3, ; Sequence 2: 6, 3, 1.5, 0.75, multiply by , ; Sequence 3: 25, 125, 625, 3125, multiply by 5,
  • Answers vary, but the point is to have students create their own equation and write terms that correspond to it.
  • 5-120.See below:
  • −4
  • 6
  • 8
  • 1040
  • x = −3, 0, 2
  • x3 − 5x − 3
  • 5-121. See below:
  • y= 23500(0.85)x, worth $2052.82
  • y= 14365112(1.12)x,population 138,570,081
  • 5-122.t(n) = −188n + 2560; 1620
  • 5-123. See below:
  • all numbers
  • 1, 2, 3, ...
  • x ≠ 0
  • 1, 2, 3, 4, ...