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

The answer is "
`\bold{(1)/(4)<k\leq 2+√(3)}`"

Explanation:

Given:

`\ S_1 = k \\\\ S_(n+1) = √(4S_n -1)`
`_(where) \ \ n \geq 1`

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

`\to √(4k-1) \leq \ k=S_1\\\\`

square the above value:

`\to k^2-4k+1 \leq 0\\\\\to k \leq 2+√(3) \ \ \ \ \ and \ \ 4k+1 >0\\\\`

`\bold{\boxed{(1)/(4)<k\leq 2+√(3)}}`