Condition for the sequence $\left\{x_n\right\}$ being monotone where $x_1>0$ and $x_{n+1}=\frac{3(1+x_n)}{5+x_n}$ [closed]

A sequence $$\left\{x_n\right\}$$ is given by $$x_1>0$$ and $$x_{n+1}=\frac{3(1+x_n)}{5+x_n}$$ for all $$n\in\mathbb{N}$$.

How can I prove that

1. The sequence $$\left\{x_n\right\}$$ is monotone increasing if $$0.
2. The sequence $$\left\{x_n\right\}$$ is monotone decreasing if $$x_1>1$$.

I cannot prove this. Should I prove this by assuming first that the sequence is monotone increasing and thereby deduce the range of $$x_1$$, or what?

closed as off-topic by rtybase, cmk, Shogun, Xander Henderson, José Carlos SantosJun 12 at 20:29

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• You neediof $0<x<1$ then $x<\frac{3(1+x)}{5+x}<1.$ And likewise, if $x>1$ then $1<\frac{3(1+x)}{5+x}<x.$ – Thomas Andrews Jun 11 at 20:51

For the case $$0, we'll prove by induction that $$0.

Induction basis: $$0.

Induction step: Assume $$0. The inequality $$0 is obvious from the recursive formula because of $$0. Additionally, we have

$$x_{n+1}=\frac{3(1+x_n)}{5+x_n}=\frac{3+3x_n}{5+x_n}<\frac{5+x_n}{5+x_n}=1.$$

Thus, the induction is complete and $$0 holds for all $$n$$.

This implies

$$x_{n+1}=\frac{3(1+x_n)}{5+x_n}>\frac{3(1+x_n)}{5+1}=\frac{1}{2}(1+x_n)>\frac{1}{2}(x_n+x_n)=x_n$$

and hence, the sequence is monotonically increasing.

The second statement for the case $$x_1>1$$ is proved analogously.

$$\frac{x_{n+1}-1}{x_{n+1}+3}=\frac{\frac{3(1+x_n)}{5+x_n}-1}{\frac{3(1+x_n)}{5+x_n}+3}=\frac{2x_n-2}{6x_n+18}=\frac{1}{3}\cdot\frac{x_n-1}{x_n+3}.$$ Thus, $$\frac{x_n-1}{x_n+3}=\frac{x_1-1}{x_1+3}\cdot\left(\frac{1}{3}\right)^{n-1}$$ or $$1-\frac{4}{x_n+3}=\frac{x_1-1}{x_1+3}\cdot\left(\frac{1}{3}\right)^{n-1}.$$ Now, we see that for $$0 the sequence $$x$$ increases (because the expression $$\frac{x_1-1}{x_1+3}\cdot\left(\frac{1}{3}\right)^{n-1}$$ increases) and for $$x_1>1$$ it decreases.

• Can the down-voter explain, why did you do it? – Michael Rozenberg Jun 12 at 7:07
• I didn't downvote, but I was downvoted too ... It looks like a revenge for the what looks like a downvoted question. I voted to close it, it lacks of details anyway. – rtybase Jun 12 at 7:43