Question 5
The sequences $(a_n)_{n\geq0}$ and $(b_n)_{n\geq0}$ are defined by
$$\begin{aligned}
a_0&=0,&a_n&=2a_{\lfloor n/2\rfloor}+n&&\text{for }n\geq1,\\
b_0&=0,&b_n&=3b_{\lfloor n/3\rfloor}+n&&\text{for }n\geq1.
\end{aligned}$$
Prove that the sequence with general term $2^{a_n}-3^{b_n}$ changes sign infinitely often.
$$\begin{aligned}
a_0&=0,&a_n&=2a_{\lfloor n/2\rfloor}+n&&\text{for }n\geq1,\\
b_0&=0,&b_n&=3b_{\lfloor n/3\rfloor}+n&&\text{for }n\geq1.
\end{aligned}$$
Prove that the sequence with general term $2^{a_n}-3^{b_n}$ changes sign infinitely often.