5.7 Arithmetic and Geometric Sequences 279 Sum of the First n Terms of a Geometric Sequence The sum of the first n terms of a geometric sequence can be determined with the following formula where a1 represents the first term and r represents the common ratio. s a r r r (1 ) 1 , 1 n n 1 = − − ≠ Example 9 Determining the Sum of a Geometric Sequence Determine the sum of the first five terms of the geometric sequence whose first term is 4 and whose common ratio is 2. Solution In this sequence, a r 4, 2, 1 = = and n 5. = Substituting these values into the formula, we get s a r r s (1 ) 1 4 1 (2) 1 2 4(1 32) 1 4( 31) 1 124 1 124 n n 1 5 5 [ ] = − − = − − = − − = − − = − − = The sum of the first five terms of the sequence is 124. The first five terms of the sequence are 4, 8, 16, 32, 64. If you add these five numbers, you will obtain the sum 124. 7 Now try Exercise 49 Example 10 Pounds and Pounds of Silver As a reward for saving his kingdom from a band of thieves, a king offered a knight one of two options. The knight’s first option was to be paid 100,000 pounds of silver all at once. The second option was to be paid over the course of a month. On the first day, he would receive one pound of silver. On the second day, he would receive two pounds of silver. On the third day, he would receive four pounds of silver, and so on, each day receiving double the amount given on the previous day. Assuming the month has 30 days, which option would provide the knight with more silver? Solution The first option pays the knight 100,000 pounds of silver. The second option pays according to the geometric sequence … 1, 2, 4, 8, 16, . In this sequence, a r 1, 2, 1 = = and n 30. = The sum of this sequence can be determined by substituting these values into the formula to obtain = − − = − − = − − = − − = s a r r s (1 ) 1 1(1 2 ) 1 2 1 1,073,741,824 1 1,073,741,823 1 1,073,741,823 n n 1 30 30 Thus, the knight would get 1,073,741,823 pounds of silver with the second option. Compared with the first option, the second would result in an additional 1,073,641,823 pounds of silver. 7 Now try Exercise 61 Learning Catalytics Keyword: Angel-SOM-5.7 (See Preface for additional details.) Instructor Resources for Section 5.7 in MyLab Math • Objective-Level Videos 5.7 • PowerPoint Lecture Slides 5.7 • MyLab Exercises and Assignments 5.7 Remigiusz Gora/Shutterstock
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