Thread: Induction Proof

Prove that $\displaystyle 2^{\frac{1}{n}} <= 1 + \frac{1}{n}$.
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?

Originally Posted by BlackBlaze

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