Exponential Functions Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook®, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform®. Copyright © 2014 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution-Non-Commercial 3.0 Unported (CC BY-NC 3.0) License (http://creativecommons.org/ licenses/by-nc/3.0/), as amended and updated by Creative Com- mons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/terms. Printed: December 7, 2014 www.ck12.org Chapter 1. Exponential Functions CHAPTER 1 Exponential Functions CHAPTER OUTLINE 1.1 Geometric Sequences 1.2 Exponential Functions 1.3 Applications of Exponential Functions 1 1.1. Geometric Sequences www.ck12.org 1.1 Geometric Sequences Learning Objectives • Identify a geometric sequence • Graph a geometric sequence. • Solve real-world problems involving geometric sequences. Introduction Consider the following question: Which would you prefer, being given one million dollars, or one penny the first day, double that penny the next day, and then double the previous day’s pennies and so on for a month? At first glance it’s easy to say "Give me the million!" But why don’t we do a few calculations to see how the other choice stacks up? You start with a penny the first day and keep doubling each day. Doubling means that we keep multiplying by 2 each day for one month (30 days). On the first day, you get 1 penny, or 20 pennies. On the second day, you get 2 pennies, or 21 pennies. On the third day, you get 4 pennies, or 22 pennies. Do you see the pattern yet? On the fourth day, you get 8 pennies, or 23 pennies. Each day, the exponent is one less than the number of that day. So on the thirtieth day, you get 229 pennies, which is 536,870,912 pennies, or $5,368,709.12. That’s a lot more than a million dollars, even just counting the amount you get on that one day! This problem is an example of a geometric sequence. In this section, we’ll find out what a geometric sequence is and how to solve problems involving geometric sequences. Identify a Geometric Sequence A geometric sequence is a sequence of numbers in which each number in the sequence is found by multiplying the previous number by a fixed amount called the common ratio. In other words, the ratio between any term and the term before it is always the same. In the previous example the common ratio was 2, as the number of pennies doubled each day. The common ratio, r, in any geometric sequence can be found by dividing any term by the preceding term. Here are some examples of geometric sequences and their common ratios. 2 www.ck12.org Chapter 1. Exponential Functions 4,16,64,256,... r = 4 (divide 16 by 4 to get 4) 15,30,60,120,... r = 2 (divide 30 by 15 to get 2) 11 11 11 11 1 11 1 11, , , , ,... r = divide by 11 to get 2 4 8 16 2 2 2 ✓ ◆ 1 1 1 1 25, 5,1, , ,... r = divide 1 by -5 to get − −5 25 −5 − 5 ✓ ◆ If we know the common ratio r, we can find the next term in the sequence just by multiplying the last term by r. Also, if there are any terms missing in the sequence, we can find them by multiplying the term before each missing term by the common ratio. Example 1 Fill is the missing terms in each geometric sequence. a) 1, ___, 25, 125, ___ b) 20, ___, 5, ___, 1.25 Solution a) First we can find the common ratio by dividing 125 by 25 to obtain r = 5. To find the first missing term, we multiply 1 by the common ratio: 1 5 = 5 · To find the second missing term, we multiply 125 by the common ratio: 125 5 = 625 · Sequence (a) becomes: 1, 5, 25, 125, 625,... b) We need to find the common ratio first, but how do we do that when we have no terms next to each other that we can divide? Well, we know that to get from 20 to 5 in the sequence we must multiply 20 by the common ratio twice: once to get to the second term in the sequence, and again to get to five. So we can say 20 r r = 5, or 20 r2 = 5. · · · Dividing both sides by 20, we get r2 = 5 = 1 , or r = 1 (because 1 1 = 1 ). 20 4 2 2 · 2 4 1 To get the first missing term, we multiply 20 by 2 and get 10. 1 To get the second missing term, we multiply 5 by 2 and get 2.5. Sequence (b) becomes: 20, 10, 5, 2.5, 1.25,... You can see that if we keep multiplying by the common ratio, we can find any term in the sequence that we want—the tenth term, the fiftieth term, the thousandth term.... However, it would be awfully tedious to keep multiplying over and over again in order to find a term that is a long way from the start. What could we do instead of just multiplying repeatedly? Let’s look at a geometric sequence that starts with the number 7 and has common ratio of 2. The 1st term is: 7 or 7 20 · We obtain the 2nd term by multiplying by 2 : 7 2 or 7 21 · · We obtain the 3rd term by multiplying by 2 again: 7 2 2 or 7 22 · · · We obtain the 4th term by multiplying by 2 again: 7 2 2 2 or 7 23 · · · · We obtain the 5th term by multiplying by 2 again: 7 2 2 2 2 or 7 24 · · · · · n 1 The nth term would be: 7 2 − · 3 1.1. Geometric Sequences www.ck12.org The nth term is 7 2n 1 because the 7 is multiplied by 2 once for the 2nd term, twice for the third term, and so on—for · − each term, one less time than the term’s place in the sequence. In general, we write a geometric sequence with n terms like this: 2 3 n 1 a,ar,ar ,ar ,...,ar − The formula for finding a specific term in a geometric sequence is: th n 1 n term in a geometric sequence: an = a1r − (a1 = first term, r = common ratio) Example 2 For each of these geometric sequences, find the eighth term in the sequence. a) 1, 2, 4,... b) 16, -8, 4, -2, 1,... Solution 2 a) First we need to find the common ratio: r = 1 = 2. The eighth term is given by the formula a = a r7 = 1 27 = 128 8 1 · In other words, to get the eighth term we start with the first term, which is 1, and then multiply by 2 seven times. 8 1 b) The common ratio is r = −16 = −2 7 1 7 ( 1)7 1 16 1 The eighth term in the sequence is a = a r = 16 − = 16 − = 16 − = − = 8 1 · 2 · 27 · 27 128 − 8 Let’s take another look at the terms in that second sequence. Notice that they alternate positive, negative, positive, negative all the way down the list. When you see this pattern, you know the common ratio is negative; multiplying by a negative number each time means that the sign of each term is opposite the sign of the previous term. Solve Real-World Problems Involving Geometric Sequences Let’s solve two application problems involving geometric sequences. Example 3 A courtier presented the Indian king with a beautiful, hand-made chessboard. The king asked what he would like in return for his gift and the courtier surprised the king by asking for one grain of rice on the first square, two grains on the second, four grains on the third, etc. The king readily agreed and asked for the rice to be brought. (From Meadows et al. 1972, via Porritt 2005) How many grains of rice does the king have to put on the last square? Solution A chessboard is an 8 8 square grid, so it contains a total of 64 squares. ⇥ The courtier asked for one grain of rice on the first square, 2 grains of rice on the second square, 4 grains of rice on the third square and so on.