9514 1404 393
Answer:
3(4/3)^2543
Step-by-step explanation:
Using logarithms base 2, we have ...
[tex](a_n)^{\log{a_n}}=(a_{n+1})^{\log{a_{n-1}}}\\\\(\log{a_n})(\log{a_n}) = (\log{a_{n-1}})(\log{a_{n+1}}) \\\\ \dfrac{\log{a_{n+1}}}{\log{a_{n}}}=\dfrac{\log{a_n}}{\log{a_{n-1}}}=\dots\dfrac{\log_2{16}}{\log_2{8}}=\dfrac{4}{3}[/tex]
That is, the ratio of each term to the previous is a constant equal to 4/3. This is the definition of a geometric sequence. This sequence has first term 3 and common ratio 4/3, so the general term is ...
[tex]\log_2 a_n=3\left(\dfrac{4}{3}\right)^{n-1}[/tex]
and the 2544th term is ...
[tex]\boxed{\log_2{a_{2544}}=3\left(\dfrac{4}{3}\right)^{2543}}[/tex]
