Monotonicity of a recursively defined sequence $s_{n+1} = \sqrt{4s_n -1}$

I am trying to answer this question:

Let $s_1 = k$ and $s_{n+1} = \sqrt{4s_n -1}$. For what values of $k$ will the sequence $s_n$ be monotone increasing?

I know the definition of monotone increasing , it's that $s_n \le s_{n+1}$ for all $n$.

But how can i prove something like that , i just need some hints?

Thanks.

• Consider $f(x) = \sqrt{4x-1}$. For which $x$ is $f(x) \geqslant x$? – Daniel Fischer Mar 23 '14 at 0:47
• mmm $\sqrt{4x-1}$ >= $x$ and then solve take square of both sides we will get $4x-1$ <= $x^2$ and then solve for x ? – kabary Mar 23 '14 at 0:53
• You swapped the sense of the inequality, but apart from that, basically yes. – Daniel Fischer Mar 23 '14 at 1:00