Question

What dose `{x_n}` converges ? when `x_1=5/2,5x_(n+1)=x_n^2+6`

Answer 1

See below.

Explanation:

This is a nonlinear difference equation. Normally it is hard to analyze convergence in this case. So we will make some basic convergence considerations.

If `x_n` converges to `x^@` then

`5x^@=(x^@)^2+6` and solving for `x^@` we have

`x^@ = {2,3}`

Checking for `x = 2+delta` we conclude that `x^@=2` is an stable attraction point for `-oo < delta < 1` and `x^@ = 3` is an inestable attraction point. Resuming, for `-oo < x_1 < 3` the sequence converges to `2` otherwise is diverges.