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

where x_n is the number of hours of sleep you get on the nth night and x₀ = 7 and x₁ = 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_n} and confirm that the terms alternately increase and decrease.

b. Show that the explicit formula

generates the terms of the sequence in part (a).

c. What is the limit of the sequence?

Transcribed Image Text:

Xn+1 || 1 (x₂ + xn−1), (Xn 2 for n = 1, 2, 3, 1, 2, 3, . . . 19 2 = 1/2 + (-) 3 3 2 Xn n , for n ≥ 0,