Induction Proof

**BlackBlaze** · Apr 6th 2010, 11:28 AM

Prove that $2^{\frac{1}{n}} <= 1 + \frac{1}{n}$ .
for n in the set of all positive numbers.

Base case is obvious.
So, tried using n = k, and adding k+1.
But then I get a load of terms and it gets too messy to handle.

Any advice?

**Drexel28** · Apr 6th 2010, 12:42 PM

Originally Posted by BlackBlaze

Prove that $2^{\frac{1}{n}} <= 1 + \frac{1}{n}$ .
for n in the set of all positive numbers.

Base case is obvious.
So, tried using n = k, and adding k+1.
But then I get a load of terms and it gets too messy to handle.

Any advice?

Hint
$2^{\frac{1}{n}}\leqslant 1+\frac{1}{n}\Leftrightarrow 2\leqslant\left(1+\frac{1}{n}\right)^n$ . That's easier to work with.

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