Calculus Problem!!
Suppose you figure out that if you oversleep one night, you undersleep the following night,
and vice-versa. The pattern follows xn+1 = (1/2)(xn + xn+1), where x0 = 7 and x1 = 6 are
the hours of sleep you get in the first two nights. What is the explicit formula?

Question

Calculus Problem!!
Suppose you figure out that if you oversleep one night, you undersleep the following night,
and vice-versa. The pattern follows xn+1 = (1/2)(xn + xn+1), where x0 = 7 and x1 = 6 are
the hours of sleep you get in the first two nights. What is the explicit formula?

anonymous · Answer

$$x _{n+1} = (1/2)(x _{n} + x _{n+1}), where x _{0} = 7 and x _{1} = 6$$

experimentx · Answer

for even:  
x(2n) = x(2n-2)
for odd:
x(2n+1) = x(2n-1)

anonymous · Answer

i have to express it as $$X _{n}=$$

experimentx · Answer

x(n) = x(n-2) .. i suppose

anonymous · Answer

$$X _{0}=7,X _{1}=6,X _{2}=6.5,X _{3}=6.25,X _{4}=6.375,X _{5}=6.3125,X _{6}=6.34375$$

experimentx · Answer

x(n+2) = 1/2 (x(n)+x(n+1))

turingtest · Answer

by "explicit formula" do you mean solve for $x_n$ ?

anonymous · Answer

yea,i think so.$$X _{n}=$$...n
and make this formula works for all the terms$$X _{0},X _{1}$$etc.

dumbcow · Answer

the limit of x_n as n-> infinity is 19/3
it alternates above and below this value 
$$\rightarrow X_{n} = \frac{19}{3}+\frac{(-1)^{n}}{3*2^{n-1}}$$

anonymous · Answer

@dumbcow  Thank you so much!!
@TuringTest @experimentX Thanks for your time too!!

turingtest · Answer

I didn't even do anything :P

anonymous · Answer

at least you looked at it,hehe^^