By considering the first three terms of the binomial expansion of \(\left[ 1+\left(\dfrac{1}{\sqrt{n}}\right) \right]^n\), where \(n\) is an integer greater than \(1\), prove that \[\left[1+\left(\frac{1}{\sqrt{n}}\right)\right]^{n-2} \geq \tfrac{1}{2}n.\]

Since \(n \geq 2\), we know that our expansion contains at least three terms.

Maybe we can obtain an inequality by throwing away the remaining terms?

What condition do these terms have to satisfy for this to be valid?

But the left-hand side has an exponent of \(n\) instead of \(n-2\). What can we do to fix this?