<s>[INST] Finding if sentence is true using induction: $\sqrt1 + \sqrt2 +\dots+\sqrt{n} \le \frac23(n+1)\sqrt{n+1}$ I have to prove
$\sqrt{1} + \sqrt{2} +...+\sqrt{n} \le  \frac{2}{3}*(n+1)\sqrt{n+1}$
by using math induction. 
First step is to prove that it works for n = 1 , which is true.
Next step is to prove it for n + 1. We can rewrite the formula using
$\sum_{i=1}^{n+1} \sqrt{i}= \sum_{i=1}^{n}\sqrt{i} + \sqrt{i+1}$
and we can substitute sum 
$\frac{2}{3}(n+1)\sqrt{n+1} +\sqrt{n+1} \le  \frac{2}{3}(n+2)\sqrt{n+2}$
we can transform the left side into 
$\sqrt{n+1}(\frac{2}{3}(n+1)+1)$
but how to I further transform the formula in order to find if the sentence is true?
Thanks for all help!
 [/INST] To finish your proof by induction we could use the fact the following inequalities are equivalent to each other
\begin{align*}
\frac23(n+1)^{3/2}+\sqrt{n+1} &\le \frac23(n+2)^{3/2}\\
\frac32\sqrt{n+1} &\le (n+2)^{3/2} - (n+1)^{3/2}\\
\frac32\sqrt{n+1} &\le (\sqrt{n+2} - \sqrt{n+1}) (n+2 + \sqrt{(n+2)(n+1)} + n+1)\\
\frac32\sqrt{n+1}(\sqrt{n+2} + \sqrt{n+1}) &\le n+2 + \sqrt{(n+2)(n+1)} + n+1\\
\frac32(n+1 + \sqrt{(n+2)(n+1)}) &\le 2n+3 + \sqrt{(n+2)(n+1)}\\
\frac12\sqrt{(n+2)(n+1)} &\le \frac{n+3}2
\end{align*}
I have:
Used $a^3-b^3=(a-b)(a^2+ab+b^2)$ for $a=\sqrt{n+2}$ $b=\sqrt{n+1}$
Multiplied both sides by $\sqrt{n+2}+\sqrt{n+1}$ to use that $(\sqrt{n+2} - \sqrt{n+1})(\sqrt{n+2} + \sqrt{n+1}) = (n+2)-(n+1)=1$.

If you already know how to integrate, you can simply use that 
$$\newcommand{\dx}{\; \mathrm{d}x}\sqrt k \le \int_{k}^{k+1} \sqrt x \dx.$$
Simply by adding these inequalities together for $k=0$ to $n$ you get
$$ \sum_{k=0}^n \sqrt n \le \int_0^{n+1} \sqrt x \dx = \frac{(n+1)^{3/2}}{3/2} = \frac23 (n+1)\sqrt{n+1}.$$
This can be also visualized by noticing that one side is are of rectangle under the curve $y=\sqrt x$ and the other is the whole area under the curve.
I will add the following - somewhat similar - picture which might help (it was taken from this post):

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