I need guidance on proving monotonicity for this sequence

[trickslapper]

The sequence is defined as x[sub:2bvzgdeo]n+1[/sub:2bvzgdeo]=3+1/x[sub:2bvzgdeo]n[/sub:2bvzgdeo]

I cannot prove monotonicity of this sequence so i split it into two subsequences. The odd terms and the even terms. I was able to prove montone decreasing for the odd terms but i'm having serious trouble proving monotonicity (increasing) for the even terms. Here are the first 3 terms of the EVEN terms of the original sequence:

x[sub:2bvzgdeo]2[/sub:2bvzgdeo]=33/10
x[sub:2bvzgdeo]4[/sub:2bvzgdeo]=360/109
x[sub:2bvzgdeo]6[/sub:2bvzgdeo]=3927/1189

So since the previous term is always less than the next term i Assume that x[sub:2bvzgdeo]n-1[/sub:2bvzgdeo]<x[sub:2bvzgdeo]n[/sub:2bvzgdeo], Now show that x[sub:2bvzgdeo]n[/sub:2bvzgdeo]<x[sub:2bvzgdeo]n+1[/sub:2bvzgdeo]

I've tried all kind of different ways and i actually end up getting a contradiction, does anyone see a way that i can prove the monotonicity of the EVEN terms of the sequence?

thanks!

 

[daon]

Re: I need guidance on proving monotonicity for this sequenc

You proved the odd's are decreasing then?

\(\displaystyle x_{2n+2} > x_{2n} \iff 3+ \frac{1}{x_{2n+1}} > 3+ \frac{1}{x_{2n-1}} \iff x_{2n-1} > x_{2n+1}\)

So (odds decreasing) if and only if (evens increasing).

 

[trickslapper]

Re: I need guidance on proving monotonicity for this sequenc

How would i prove it assuming that i hadn't proved the odds were montone decreasing? I guess i should have been more specific sorry!

 

[daon]

Re: I need guidance on proving monotonicity for this sequenc

from before...

\(\displaystyle x_{2n+2} > x_{2n} \iff \frac{1}{x_{2n+1}} > \frac{1}{x_{2n-1}} \iff x_{2n-1} > x_{2n+1}\)

then

\(\displaystyle x_{2n-1} > x_{2n+1} \iff 3+\frac{1}{x_{2n-2}} > 3+\frac{1}{x_{2n}} \iff x_{2n} > x_{2n-2}\)

Assuming of course, you proved x_n are always positive