# Math Help - Inequality

1. ## Inequality

Show that $1<\sqrt[n]{a}<\frac {n+(a-1)}n$ if $a>1$ and $n\in\mathbb{N}\setminus\{1\}$.

2. That $\sqrt[n]{a}>1$ is easy to see. For the second inequality, $\sqrt[n]{a}<\frac {n+(a-1)}n\iff a<(1+(a-1)/n)^n$. The latter can be shown using the binomial theorem, according to which $(1+x)^n=1+nx+\dots>1+x$.

In fact, the whole power of the theorem is not needed. One can write $(1+x)^n$ as $(1+x)\cdot(1+x)\cdot\;\dots\;\cdot(1+x)$. When performing the multiplication, we have to pick either 1 or x from each factor, and consider and add all such variants. There is only one way to choose 1 from every factor; all other terms will have x in them. Then we can choose x from the first factor and 1's from all the rest, or x from the second factor and 1's from all the rest, etc. In the sum, this will give us $nx$. All the rest terms in the sum will have at leat $x^2$.