Prove that the inequality:$$\sqrt{1}+\sqrt{2} +\sqrt{3}+\cdots+\sqrt{n}\geq\left(\frac{n\sqrt{n}+1}{2}\right)$$ by mathematical induction, if $n\in\mathbb{N}$.
What I tried :
Step 1:
$p(1)$ : $\sqrt{1}\geq\left(\frac{1\times\sqrt{1}+1}{2}\right) \implies 1=1$ .
Step 2: $k\in\mathbb{N}$.
$p(k)$: $\sqrt{1}+\sqrt{2} +\sqrt{3}+\cdots+\sqrt{k}\geq\left(\frac{k\sqrt{k}+1}{2}\right)$ which is assumed.
$p(k+1)$: $\sqrt{1}+\sqrt{2} +\sqrt{3}+\cdots+\sqrt{k}+\sqrt{k+1}\geq\left(\frac{(k+1)\sqrt{k+1}+1}{2}\right)$
$\sqrt{1}+\sqrt{2} +\sqrt{3}+\cdots+\sqrt{k}\geq\left(\frac{k\sqrt{k}+1}{2}\right)$ this part is $p(k)$, so I am going to add $\sqrt{k+1}$ on both sides:
$$\left(\frac{k\sqrt{k}+1}{2}\right)+ \sqrt{k+1}$$
From here I don’t know how to continue and I’m not sure if the adding part is even helpful.
I hope the question is understandable. Thanks !