Question

Let s1 = k and define sn+1 = √4sn − 1 for n ≥ 1. Determine for what values of k the sequence (sn) will be monotone increasing and for what values of k it will be monotone decreasing.

Answer 1

Answer:

The answer is "$\bold{\frac{1}{4}$"

Step-by-step explanation:

Given:

$\ S_1 = k \\\\ S_{n+1} = \sqrt{4S_n -1}$   $_{where} \ \ n \geq 1$

In the above-given value, $S_n$ is required for the monotone decreasing so, $S_2 :$

$\to \sqrt{4k-1} \leq \ k=S_1$

square the above value:

$\to k^2-4k+1 \leq 0\\\\\to k \leq 2+\sqrt{3} \ \ \ \ \ and \ \ 4k+1 >0$

$\bold{\boxed{\frac{1}{4}$