Question 140
Decide whether for every sequence $\left(a_{n}\right)$ of positive real numbers,
$$
\frac{3^{a_{1}}+3^{a_{2}}+\cdots+3^{a_{n}}}{\left(2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{n}}\right)^{2}}\lt \frac{1}{2024}
$$
is true for at least one positive integer $n$.
$$
\frac{3^{a_{1}}+3^{a_{2}}+\cdots+3^{a_{n}}}{\left(2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{n}}\right)^{2}}\lt \frac{1}{2024}
$$
is true for at least one positive integer $n$.