# Mathematical Induction Problem

1. Nov 3, 2005

### MrBailey

Hi, all.
I'm working on some proof by induction problems. While I understand the concept, this one threw me for a loop.
Let $$x_1=\sqrt{2}$$ and $$x_{n+1}=\sqrt{2+x_n}$$
Show that $$x_n < x_{n+1}$$
I'd greatly appreciate help with this.
Thanks,
bailey

2. Nov 3, 2005

### Gale

sure you need to use induction? i would show that the stuff under the radical for $$x_{n+1}> x_n$$ we know this because $$x_n>0$$. and 2 plus some other positive number will always be greater than two, and therefore the sq rt of that sum will be greater eh?

3. Nov 3, 2005

### AKG

Uh, you certainly need induction. If xn = 98, then xn+1 = (2 + 98)1/2 = 1001/2 = 10 < 98 = xn.

Show that x1 < x2
Assume that xk < xk+1
Use this to prove that xk+1 < xk+2
Write out xk+1 and xk+2 in terms of xk. Then there xk+1 < xk+2 will follow immediately from xk < xk+1 as long as you know that the function f defined by f(a) = a1/2 is an increasing function.

4. Nov 4, 2005

### MrBailey

thanks...much clearer now

Bailey