8.4 Start Thinking. 8.4 Warm Up. 8.4 Cumulative Review Warm Up. List the first 10 terms of the geometric sequence = ( ) n

Transcription

1 . Start Thinking List the first 0 terms of the geometric sequence ( ) 0 Then find the value of ( ) the value of ( ) n n a n n and make a conjecture about. Warm Up Find the sum.. n. n.. n. ( ) n 00 n 6. ( ) n. Cumulative Review Warm Up Graph the function. State the domain and range.. f ( x) x. f ( x) x. f ( x) x. f ( x) x x. f ( x) x x 6. f ( x) x x

2 Name Date. Practice A In Exercises and, consider the infinite geometric series. Find the partial sums Sn for n,,,, and. Then describe what happens to S n as n increases In Exercises 6, find the sum of the infinite geometric series, if it exists. n. ( ). ( ) n Describe and correct the error in finding the sum of the infinite geometric series. F B not have a sum.. You push your younger sister on a swing one time and then allow your sister to swing freely. On the first swing, your sister travels a distance of feet. On each successive swing, your sister travels 0% of the distance of the previous swing. What is the total distance your sister swings? In Exercises 9, write the repeating decimal as a fraction in simplest form A company had a profit of $00,000 in its first year. Since then, the company s profit has decreased by 6% each year. Assuming this trend continues, what is the total profit the company can make over the course of its lifetime? Algebra Copyright Big Ideas Learning, LLC

3 Name Date. Practice B In Exercises and, consider the infinite geometric series. Find the partial sums Sn for n,,,, and. Then describe what happens to S n as n increases In Exercises 6, find the sum of the infinite geometric series, if it exists. n. ( ). ( ) n Describe and correct the error in finding the sum of the infinite geometric series. F. You are going for a -mile run. You know that you can run half the distance, and you successfully run miles. There are miles to go, and you know that you can run half that distance. You successfully run that next mile. Now there is mile to go, and you know that you can run half that distance. You successfully run that next half mile. This process continues. Will you cover the miles over the course of your run? Explain your answer. In Exercises 9, write the repeating decimal as a fraction in simplest form A radio station has a daily contest in which a random listener is asked a trivia question. On the first day, the station gives $00 to the first listener who answers correctly. On each successive day, the winner receives 9% of the winnings from the previous day. What is the total amount of prize money the radio station gives away during the contest?

4 Name Date. Enrichment and Extension Finding Sums of Infinite Geometric Series Example: For what values of x does the following infinite series converge? ( x ) ( x ) ( x ) Solution: This is an infinite geometric series with r x. By the Sum of an Infinite Geometric Series Theorem, the series converges when r < ; that is, when x <, or < x <. This interval < x < for which the series converges is called the interval of convergence for the series. In Exercises 6, find (a) the interval of convergence and (b) the sum, expressed in terms of x.. 6 x x x. x 9x. ( x ) ( x ) ( x ). ( x ) ( x ) ( x ) x x x x x x 6. Show that the series π if x nπ. 6 sin x sin x sin x converges to tan x. Show that the series 6 tan x tan x tan x converges to sin x π π if < x <. Are there other values of x for which the series converges? 9. Explain why there is no infinite geometric series with first term 0 and sum. 6 Algebra Copyright Big Ideas Learning, LLC

5 Name Date. Puzzle Time What Kind Of Dog Does Dracula Have? Write the letter of each answer in the box containing the exercise number. Find the sum of the infinite geometric series, if it exists i i i 6 Answers H. N. 9 B. O. D. O. 6.. i i i i O. no sum U. L. 9. i 9 D. 0. i 6 i 6 9 0