Review question

# Can we find an inequality from the expansion of $[ 1+1/\sqrt{n}]^n$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6770

## Suggestion

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?