Sleep model After many nights of observation, you notice that if you oversleep one night, you tend to undersleep the following night, and vice versa. This pattern of compensation is described by the relationship 1 Xn+1 =(*, + x,,-1), for n = 1, 2, 3, . . . , where x, is the number of hours of sleep you get on the nth night, and x, = 7 and x, = 6 are the number of hours of sleep on the first two nights, respectively. a. Write out the first six terms of the sequence {x,} and confirm that the terms alternately increase and decrease. b. Show that the explicit formula 19 for n > 0, 3 3 generates the terms of the sequence in part (a). c. Assume the limit of the sequence exists. What is the limit of the sequence?
Sleep model After many nights of observation, you notice that if you oversleep one night, you tend to undersleep the following night, and vice versa. This pattern of compensation is described by the relationship 1 Xn+1 =(*, + x,,-1), for n = 1, 2, 3, . . . , where x, is the number of hours of sleep you get on the nth night, and x, = 7 and x, = 6 are the number of hours of sleep on the first two nights, respectively. a. Write out the first six terms of the sequence {x,} and confirm that the terms alternately increase and decrease. b. Show that the explicit formula 19 for n > 0, 3 3 generates the terms of the sequence in part (a). c. Assume the limit of the sequence exists. What is the limit of the sequence?
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:Sleep model After many nights of observation, you notice that
if you oversleep one night, you tend to undersleep the following
night, and vice versa. This pattern of compensation is described
by the relationship
1
Xn+1 =(*, + x,,-1), for n = 1, 2, 3, . . . ,
where x, is the number of hours of sleep you get on the nth night,
and x, = 7 and x, = 6 are the number of hours of sleep on the
first two nights, respectively.
a. Write out the first six terms of the sequence {x,} and confirm
that the terms alternately increase and decrease.
b. Show that the explicit formula
19
for n > 0,
3
3
generates the terms of the sequence in part (a).
c. Assume the limit of the sequence exists. What is the limit of
the sequence?
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