# Thread: Show by mathematical induction:

1. Of course Thanks!

2. Originally Posted by jenkki
Yes Don't know how to write sub/sup text...
I don't think induction is necessary. To show that this is always positive, show that the numerator and denominator are always the same sign (I can't see any reason why either of them would ever be negative...)

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3. No, it's not necessary.
However, the objective is to learn and practice the technique of Proof By Induction.

4. Originally Posted by Prove It
I don't think induction is necessary. To show that this is always positive, show that the numerator and denominator are always the same sign (I can't see any reason why either of them would ever be negative...)
??The denominator is just $\displaystyle 2a_n$. Saying that "I can't see any reason why either of them would ever be negative" is exactly the same as saying "I can't see any reason why $\displaystyle a_n$ wouldn't be positive"- and that does NOT constitute a proof that it is!

Indeed, since the numerator, $\displaystyle a_n^2+ 2$ is obviously always positive, saying "show that the numerator and denominator are always the same sign" is exactly the same as saying "show that $\displaystyle a_n$ is always positive" which is exactly what the original problem says!

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