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?


  • 3
    $\begingroup$ Consider $f(x) = \sqrt{4x-1}$. For which $x$ is $f(x) \geqslant x$? $\endgroup$ – Daniel Fischer Mar 23 '14 at 0:47
  • $\begingroup$ 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 ? $\endgroup$ – kabary Mar 23 '14 at 0:53
  • $\begingroup$ You swapped the sense of the inequality, but apart from that, basically yes. $\endgroup$ – Daniel Fischer Mar 23 '14 at 1:00

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