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1. The problem statement, all variables and given/known data

Let x be any real number in the interval (0,1). Prove that for any natural number n greater or equal to 2 it is true that

(1+x^2)^n is greater or equal to (1+x^n)^2

(^ means exponent)

2. Relevant equations

3. The attempt at a solution

Ok well I have been attempting to do this problem for a couple days and it has come to a point when I am not getting any further.:grumpy:

Here is what I have so far:

I believe that I should be doing a proof by Induction.

Proof:

Let n be a natural number greater than or equal to 2. Let 0 < x <1 and let p(n): (1+x^2) >(or equal to) (1+x^n)^2

Base case: WWTS p(2) is true.

p(2): (1+x^2)^2 >(or equal to) (1+x^2)^2. As (1+x^2)^2 = (1+x^2)^2, p(2) is true.

Inductive step: Assume p(n): (1+x^2)^n >(or equal to) (1+x^n)^2 is true. WWTS p(n+1): (1+x^2)^(n+1) >(or equal to) (1+x^(n+1))^2 is true.

I guess this is basically where I get stuck. I know that I have to use my assumption p(n) to try to get to p(n+1). I have tried adding different things to each side, multiplying each side by different things but nothing seems to give me what I want.

I recognize that to get the left sides of the equations to be the same I need to multiply (1+x^2)^n by (1+x^2). And in order to get the right sides to look the same I need to first multiply each out to get rid of the ^2 (just so it is easier to see what is needed)....

But I am seriously stuck. Any help would be greatly appreciated.

Thanks

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Induction proof please help

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