taiwan-gsat 2024 Q11

taiwan-gsat · Other · ast__math-a 6 marks Sequences and series, recurrence and convergence Closed-form expression derivation

Let the real numbers $a_{1}, a_{2}, \ldots, a_{9}$ form an arithmetic sequence with common difference 2, where $a_{1} \neq 0$ and $a_{3} > 0$. If $\log_{2} a_{3}$, $\log_{2} b$, $\log_{2} a_{9}$ form an arithmetic sequence in order, where $b$ is one of $a_{4}, a_{5}, a_{6}, a_{7}, a_{8}$, then $a_{9} = $ . (Express as a simplified fraction)

Let the real numbers $a_{1}, a_{2}, \ldots, a_{9}$ form an arithmetic sequence with common difference 2, where $a_{1} \neq 0$ and $a_{3} > 0$. If $\log_{2} a_{3}$, $\log_{2} b$, $\log_{2} a_{9}$ form an arithmetic sequence in order, where $b$ is one of $a_{4}, a_{5}, a_{6}, a_{7}, a_{8}$, then $a_{9} = $ \underline{\hspace{2cm}}.\\
(Express as a simplified fraction)

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