# Thread: Show the sequence is decreasing

1. ## Show the sequence is decreasing

I have attached a picture of the question. How would you go about showing this? Thanks.

2. Originally Posted by universalsandbox
I have attached a picture of the question. How would you go about showing this? Thanks.

Is/are there an extra condition/s that is/are missing here?

RonL

3. Originally Posted by CaptainBlack
Is/are there an extra condition/s that is/are missing here?

RonL
Ex:

let a=2, xknot = 1

then

x(sub1) = (1/2)(xknot + 2/xknot) = 3/2

x(sub2) = (1/2)( x(sub1) + 2/(xsub1) ) = 17/2 etc.

4. I work out that if $x_n > \frac{a}{x_n}$,

$x_n + \frac{a}{x_n} > \frac{2a}{x_n}$

$0.5\left(x_n + \frac{a}{x_n}\right) > \frac{a}{x_n}$

$x_{n+1} > \frac{a}{x_n}$.

As long as $x_n > \frac{a}{x_n}$, then the function will be decreasing, but I can't find a way to show that statement inductively.

5. Originally Posted by universalsandbox
Ex:

let a=2, xknot = 1

then

x(sub1) = (1/2)(xknot + 2/xknot) = 3/2

x(sub2) = (1/2)( x(sub1) + 2/(xsub1) ) = 17/2 etc.
Compare what happens when $x_0>\sqrt{a}$ with what happens when $x_0<\sqrt{a}$ with what happens when $x_0=\sqrt{a}.$

RonL