<s>[INST] How would you prove that $2^{50} < 3^{33}$ without directly calculating the values Could you generalise the question and get something along the lines of $n^{50} < (n+1)^{33}$ ?
 [/INST] \begin{align}
3^2=2^3+1\quad&\Leftrightarrow\quad \underbrace{(3^2)^{17}}_{3^{34}}=(2^3+1)^{17}=[\text{binomial}]=2^{51}+17\cdot 2^{48}+\text{positive}\quad\Rightarrow\quad \\
&\Rightarrow\quad 3\cdot 3^{33}>2\cdot 2^{50}+\frac{17}{4}\cdot 2^{50}>3\cdot 2^{50}. 
\end{align}
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