# Thread: prove sequence is decreasing using PMI

1. ## prove sequence is decreasing using PMI

*x_n indicates x sub n
*>= indicates greater than or equal to

Let x_1 >= 2 and x_(n+1) = 1+ sqrt(x_n - 1)

I can prove that x_n>=x_n+1 --> x_n+1 >= x_n+2 but when I try to do the base case (or what I think is the base case), I can't show x_1>=x>2:

Please help me show that x_1 >= x_2 for my base case.

Unless I am doing this completely wrong. This is my first induction problem using sequences, so I have no clue. Thanks!

2. A similar problem [posted by the same user...] has been solved recently in...

http://www.mathhelpforum.com/math-he...it-170087.html

The difference equation can be written as...

$\displaystyle \displaystyle \Delta_{n}= x_{n+1}-x_{n} = 1+\sqrt{x_{n}-1} - x_{n} = f(x_{n})$ (1)

Following the trace of the post I have indicated You should be able to...

a) find the 'attractive fixed point' $\displaystyle x_{0}$

b) find the range of values of $\displaystyle x_{1}$ for which the sequence tends to $\displaystyle x_{0}$

c) extablish that the convergence is monotonic or not

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$