Question 3
Decide whether for every sequence $ (a_{n}) $ of positive real numbers,
$$ \frac{3^{a_{1}}+3^{a_{2}}+\cdots+3^{a_{n}}}{(2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{n}})^{2}} \lt \frac{1}{2024} $$
is true for at least one positive integer n.
Comment. The question can be asked in several forms, as follows:
(i) students could be asked, as above, to show the existence of such an n;
(ii) students could be asked to show that this happens for all sufficiently large n;
(iii) students could be given a concrete positive integer N and asked to show it for all $n\gt N$.
The solutions below provide varying bounds for N.
$$ \frac{3^{a_{1}}+3^{a_{2}}+\cdots+3^{a_{n}}}{(2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{n}})^{2}} \lt \frac{1}{2024} $$
is true for at least one positive integer n.
Comment. The question can be asked in several forms, as follows:
(i) students could be asked, as above, to show the existence of such an n;
(ii) students could be asked to show that this happens for all sufficiently large n;
(iii) students could be given a concrete positive integer N and asked to show it for all $n\gt N$.
The solutions below provide varying bounds for N.