Proving inequality (induction?) Let $a>0$, $n,p \in \mathbb{N}$, $p<n$
$\sqrt[\leftroot{-1}\uproot{2}n]{a^p} \leq 1+\frac{p}{n} (a-1)$
I tried to do it by induction, with the first step for $a=2, n=2, p=1$.
$\sqrt{2^1} \leq 1+ \frac{1}{2}(2-1)$. However I do not know how to go on with the induction. Would appreciate help

Apply the AM-GM inequality:
$$
(x_1\cdot x_2\cdots x_n)^{\frac{1}{n}}\leq \frac{x_1+x_2+\ldots+x_n}{n}
$$
to $\Big(\underbrace{a\cdots a}_{\text{p times}}  \cdot  \underbrace{1\cdots 1}_{\text{(n-p) times}} \Big)$ gives
$$
a^{\frac{p}{n}}\leq \frac{pa+n-p}{n}=\frac{p}{n}a+\frac{n-p}{n}=1+\frac{p}{n}(a-1).
$$
