<s>[INST] Prove $5^n + 3^n - 2^{2n+1} > 0$ by induction I am not sure how to deal with the $-2^{2n+1}$ term.
I did the basis proof for n=1
I am stuck at this step:
$$
5^{k+1}+3^{k+1}-2^{2(k+1)+1} = 5\cdot 5^k + 3 \cdot 3^k -2^3 \cdot 2^{2k}
$$
Any advice guys?
 [/INST] Note that
$$
\begin{align}
5^{k+1}+3^{k+1}-2^{2(k+1)+1}
&=5\cdot5^k+3\cdot3^k-4\cdot2^{2k+1}\\
&=\left(5^k-3^k\right)+4\left(5^k+3^k-2^{2k+1}\right)
\end{align}
$$
</s>