<s>[INST] How to prove $n^{3/2} \le 2^n/n$ for all $n\geq 8$ via Induction? How to prove $n^{3/2} \le 2^n/n$ for all $n \geq 8$
My attempt:
$\textbf{Base case}$ is $n_0=8$: $\quad   \left(8^{3/2}=4\right)\leq \left(32=\frac{2^8}{8}\right) \checkmark$
$\textbf{Induction hypothesis}$: $\exists n\in \mathbb{N}_{\geq 8}:(n+1)^{3/2}\leq \frac{2^{n+1}}{n+1}$
$\textbf{Inductive step}$: 
$$
\begin{gather}
(n+1)^{3/2}\leq \frac{2^{n+1}}{n+1} \end{gather} \\
  \iff (n+1)^{5/3}\leq 2^{n+1} \\
  \iff (n+1)^{5/3}\leq 2^n \cdot 2 \quad |\uparrow ^{3/5} \\
  \iff n+1\leq 2^{3n/5}\cdot 2^{3/5}\ $$
This, sadly, doesn't get me anywhere, any ideas?
 [/INST] Consider function $f(x)=4^x-x^5$. Now we have to prove that $f(x)>0$ for $x\geq8$ using induction.
Let start with $f(8)=4^8-8^5=2^{16}-2^{15}\gt0$
Now, for any $x=k\geq8$ let us assume $f(k)=4^k-k^5\geq0$
Now put $x=k+1$, then $f(k+1)=4^{k+1}-(k+1)^5=4\times4^k-(k+1)^5$
$f(k+1)=[4^k-k^5]+[4^k-5k^4]+[4^k-10k^3]+[4^k-10k^2-5k^-1]$
$f(k+1)=[4^k-k^5]+[4^k-\frac{5}{k}k^5]+[4^k-\frac{10}{k^2}k^5]+[4^k-(\frac{10}{k^3}+\frac{5}{k^4}+\frac{1}{k^5})k^5]$
$f(k+1)=t_1+t_2+t_3+t_4$  (Assume all one square bracket as a single term)
since we are assuming $t_1\geq0$ and in other terms, the quotient of $k^5$ in $t_2,t_3,t_4$ is $\lt1$ which makes positive part more dominant.
Hence, all the terms are positive giving $f(k+1)\geq0$
Now, by theorem of Induction, $f(k)\geq0$ for $x\geq8$.
Hence Proved!
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