Correction

**simplysparklers** · May 18th 2008, 06:58 AM

Hi guys
Can anyone put me right on this? I keep coming up with the opposite answer.

$a_{1}=0, a_{n+1}=\frac{3a_{n}+1}{a_{n}+3}$
$(n\geq{1})$
Show that $a_{n}$ is monotonic increasing.

Anyway I do it, I keep finding $a_{n}$ is monotonic decreasing. I've tried doing it as $a_{n+1}-a_{n}\geq{0}$ , and I've gone very simple and just substituted different random values in for $a_{n}$ to see what kind of resuts it would yield, but it always ends up as $a_{n+1}\leq{a_{n}}$ . Can anyone show me where I'm going wrong please??

Thanks,
Jo

**Isomorphism** · May 18th 2008, 07:24 AM

Originally Posted by simplysparklers

Hi guys
Can anyone put me right on this? I keep coming up with the opposite answer.

$a_{1}=0, a_{n+1}=\frac{3a_{n}+1}{a_{n}+3}$
$(n\geq{1})$
Show that $a_{n}$ is monotonic increasing.

Anyway I do it, I keep finding $a_{n}$ is monotonic decreasing. I've tried doing it as $a_{n+1}-a_{n}\geq{0}$ , and I've gone very simple and just substituted different random values in for $a_{n}$ to see what kind of resuts it would yield, but it always ends up as $a_{n+1}\leq{a_{n}}$ . Can anyone show me where I'm going wrong please??

Thanks,
Jo

$a_{n+1} - a_{n} = \frac{3a_n + 1}{a_n + 3} - a_n = \frac{1 - a_n ^2}{3 + a_n}$

In the previous thread we proved that $0 \leq a_n < 1$ .This implies $1 - a_n ^2 > 0, 3+a_n > 0$ . These two together imply $a_{n+1} - a_{n} > 0$

**earboth** · May 18th 2008, 07:38 AM

Originally Posted by simplysparklers

Hi guys
Can anyone put me right on this? I keep coming up with the opposite answer.

$a_{1}=0, a_{n+1}=\frac{3a_{n}+1}{a_{n}+3}$
$(n\geq{1})$
Show that $a_{n}$ is monotonic increasing.

...

1.
$<br /> a_{n+1}-a_n=\frac{3 \cdot \frac{3a_n+1}{a_n+3}+1}{\frac{3a_n+1}{a_n+3}+3}- \frac{3a_n+1}{a_n+3} = \frac{6a_n+13-3(a_n)^2}{(a_n+3)(3a_n+5)}$

Since $a_n<1$ for all n this fraction is positiv thus the sequence is monotonic increasing

2. I've attached a screenshot of a spreadsheet calculating $a_n$

3. What are you doing in Bamberg?

**simplysparklers** · May 18th 2008, 07:49 AM

Thanks Earboth!!
& I'm on my Erasmus year in Bamberg!I study German and Maths, and I have maths work sent over from my home uni to be done!Awful book to work from, and no lecturer to help me, that's why I'm so grateful I found you MHF guys!!

Whereabouts are you?

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