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HW2 even solutions

HW2 even solutions - Homework 2 Solutions 9.2.6 No Ratio...

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Homework 2 Solutions 9.2.6 No. Ratio between successive terms is not constant: 2 x 2 x = 2 x , while 3 x 3 2 x 2 = 3 2 x . 9.2.8 Yes, a = y 2 , ratio of successive terms is y . 9.2.16 The series has 10 terms. The first term is a = 0 . 2 and the constant ratio is x = 0 . 1 , so Sum = 0 . 2(1 - x 10 ) (1 - x ) = 0 . 2(1 - (0 . 1) 10 ) 0 . 9 = 0 . 222 . 9.2.18 - 2 + 1 - 1 2 + 1 4 - 1 8 + 1 16 - · · · = n =0 ( - 2) - 1 2 n , a geometric series. Let a = - 2 and x = - 1 / 2 . Then n =0 ( - 2) - 1 2 n = a 1 - x = - 2 1 - ( - 1 / 2) = - 4 3 . 9.2.20 Using the formula for the sum of a finite geometric series, 20 n =4 1 3 n = 1 3 4 + 1 3 5 + · · · + 1 3 20 = 1 3 4 1 + 1 3 + 1 3 2 + · · · + 1 3 16 = (1 / 3) 4 (1 - (1 / 3) 17 ) 1 - (1 / 3) = 3 17 - 1 2 · 3 20 . 9.2.29 (b) The first drop from 10 feet takes (1 / 4) 10 seconds. The first full bounce (to 10 · (3 / 4) feet) takes (1 / 4) 10 · (3 / 4) seconds to rise, therefore the same time to come down. Thus, the full bounce, up and down, takes 2(1 / 4) 10 · (3 / 4) seconds. The next full bounce takes 2(1 / 4)10 · (3 / 4) 2 = 2(1 / 4) 10( 3 / 4) 2 seconds. The n th bounce takes 2(1 / 4) 10( 3 / 4) n seconds. Therefore, the total amount of time is given by 1 4 10 + 2 4 10 3 4 + 2 4 10 3
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