39. Communications Many companies use a telephone chain to notify employees of a closing due to bad weather. Suppose a company's CEO (Chief Executive Officer) calls four people. Then each of these people calls four others, and so on.
    1. Make a diagram to show the first three stages in the telephone chain. How many calls are made at each stage?
    2. Write the series that represents the total number of calls made through the first six stages.
    3. How many employees have been notified after stage six?

40. Graphing Calculator The graph models the sum of the first n terms in the geometric series with a 1 = 20 eh sub 1 , equals 20  and r = 0.9 . r equals 0.9 .

    
    Image Long Description

    1. Write the first four sums of the series.
    2. Use the graph to evaluate the series to the 47th term.
    3. Write and evaluate the formula for the sum of the series.
    4. Graph the formula using the window values shown. Use the graph to verify your answer to part (b).

Evaluate each infinite series that has a sum.

41. ∑ n = 1 ∞ ( 1 5 ) n − 1 sum , from , n equals 1 , to , infinity , of . open , 1 fifth , close super n minus 1 end super
42. ∑ n = 1 ∞ 3 ( 1 4 ) n − 1 sum , from , n equals 1 , to , infinity , of . 3 . open , 1 fourth , close super n minus 1 end super
43. ∑ n = 1 ∞ ( − 1 3 ) n − 1 sum , from , n equals 1 , to , infinity , of . open , negative , 1 third , close super n minus 1 end super
44. ∑ n = 1 ∞ 7 ( 2 ) n − 1 sum , from , n equals 1 , to , infinity , of . 7 . open 2 close super n minus 1 end super
45. ∑ n = 1 ∞ ( − 0.2 ) n − 1 sum , from , n equals 1 , to , infinity , of . open , negative 0.2 , close super n minus 1 end super
46. Open-Ended Write an infinite geometric series that converges to 3. Use the formula to evaluate the series.
47. Reasoning Find the specified value for each infinite geometric series.
    1. a 1 = 12 , S = 96 ; eh sub 1 , equals 12 . comma , s equals 96 , semicolon  find r
    2. S = 12 , r = 1 6 , s equals 12 comma r equals , 1 sixth . comma  find a 1 eh sub 1
48. Writing Suppose you are to receive an allowance each week for the next 26 weeks. Would you rather receive (a) $1000 per week or (b) $.02 the first week, $.04 the second week, $.08 the third week, and so on for the 26 weeks? Justify your answer.
49. The sum of an infinite geometric series is twice its first term.
    1. Error Analysis A student says the common ratio of the series is 3 2 . 3 halves , .  What is the student's error?
    2. Find the common ratio of the series.
50. Physics Because of friction and air resistance, each swing of a pendulum is a little shorter than the previous one. The lengths of the swings form a geometric sequence. Suppose the first swing of a pendulum has a length of 100 cm and the return swing is 99 cm.
    1. On which swing will the arc first have a length less than 50 cm?
    2. What is the total distance traveled by the pendulum when it comes to rest?
51. Where did the formula for summing finite geometric series come from? Suppose the geometric series has first term a 1 eh sub 1  and constant ratio r, so that S n = a 1 + a 1 r + a 1 r 2 + ⋯ + a 1 r n − 1 . s sub n , equals . eh sub 1 , plus , eh sub 1 , r plus , eh sub 1 , r squared , plus math axis ellipsis plus , eh sub 1 . r super n minus 1 end super . .
    1. Show that r S n = a 1 r + a 1 r 2 + a 1 r 3 + ⋯ + a 1 r n . r s sub n , equals , eh sub 1 , r plus , eh sub 1 , r squared , plus , eh sub 1 , r cubed , plus math axis ellipsis plus , eh sub 1 , r to the n , .
    2. Use part (a) to show that S n − r S n = a 1 − a 1 r n . s sub n , minus , r s sub n , equals , eh sub 1 , minus , eh sub 1 , r to the n , .
    3. Use part (b) to show that S n = a 1 − a 1 r n 1 − r = a 1 ( 1 − r n ) 1 − r . s sub n , equals . fraction eh sub 1 , minus , eh sub 1 , r to the n , over 1 minus r end fraction . equals . fraction eh sub 1 . open . 1 minus , r to the n . close , over 1 minus r end fraction . .

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