so x(1) := 8, and x(n + 1) = 1/2(X(n)) + 2. show that it is bounded and monotone, and find the limit.(adsbygoogle = window.adsbygoogle || []).push({});

so I claim that it is decreasing, i.e, X(n)> x(n+1).

by induction:

n = 1 implies 8 > 6. this checks.

assume K is true, try K + 1. so need to show X(k+1) > x(K+2). we know that X(K+2) = 1/2(x(k+1)) + 2, in othe words, need to show 1/2X(k+1) + 2 < (x(K+1)). bring x(k+1) to other side, then 2 < x(K+1) - 1/2x(k+1).

simplify and get 4 <= x(k+1).

so now, I know that x(k+1) is less then or equal to 4. this shows that it is bounded by 4 right? I was trying to show that it was decreasing...but I showed this instead. Does this assume that it is decreasing? and how?

and the final question. how do I show that the limit is 4?'

and question 2:

x1 >1, and x(n+1):= 2 - 1/x(n). same thing. now since x 1 is not specifically a number, I dotn know how to approach it.

so if I just do soem computations: I know that x1 = something greater then 1, so if I want x2, that is := 2 - 1/(x1), and x3 := 2 - 1/(x2). I'm gonna guess that this will haev a bound of 2. now, since x1 is greater then 1, and it has a bound of 2. upper or lower, my head hurts, so I"m gonna take a break and tackle tihs tommorow...but.....let me think about it.

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# Find limit by induction

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