I tried to prove using mathematical induction the following inequality
$(1+\frac{3}{\sqrt{n}})^n>n$
I am not going to write the first two steps, just the third one for $n=k+1$, I got
$(1+\frac{3}{\sqrt{k+1}})^{k+1}>(1+\frac{3}{\sqrt{k+1}})^{k}(1+\frac{3}{\sqrt{k+1}})$
And everytime I find something that is smaller than the last result so I can not see how to use the second step from the induction, $(1+\frac{3}{\sqrt{k}})^{k}>k$, to show the inequality. I tried to use somehow Bernoulli inequality but i got nothing.
Thanks a lot, if you can give me just a little hint or instruction.