- Since $\alpha$, $7$, $\beta$ is a geometric sequence, then we have
$\begin{array}{rcl}
\dfrac{7}{\alpha} & = & \dfrac{\beta}{7} \\
49 & = & \alpha \beta \\
\log_7 49 & = & \log_7 (\alpha\beta) \\
\log_7 7^2 & = & \log_7 \alpha +\log_7\beta \\
\log_7 \alpha & = & 2 -\log_7 \beta
\end{array}$ - For $\log_\beta \alpha$, $\log_7\beta$, $\log_\alpha \beta$ form a arithmetic sequence,
$\begin{array}{rcl}
\log_7 \beta -\log_\beta \alpha & = & \log_\alpha \beta -\log_7\beta \\
\log_7 \beta -\dfrac{\log_7 \alpha}{\log_7 \beta} & = & \dfrac{\log_7 \beta}{\log_7\alpha} -\log_7\beta \\
\log_7\alpha(\log_7 \beta)^2-(\log_7\alpha)^2 & = & (\log_7\beta)^2 -\log_7\alpha(\log_7 \beta)^2 \\
(2 -\log_7 \beta)(\log_7 \beta)^2 -(2 -\log_7 \beta)^2 & = & (\log_7 \beta)^2 -(2 -\log_7 \beta)(\log_7\beta)^2 \\
(2 -\log_7 \beta)[(\log_7\beta)^2 -2 +\log_7\beta] & = & (\log_7 \beta)^2 (1-2+\log_7\beta) \\
(2-\log_7 \beta)(\log_7\beta+2)(\log_7\beta-1) & = & (\log_7\beta)^2(\log_7\beta -1) \\
(2-\log_7 \beta)(\log_7\beta+2) & = & (\log_7\beta)^2 \\
4-(\log_7\beta)^2 & = & (\log_7\beta)^2 \\
(\log_7\beta)^2 & = & 2
\end{array}$$\therefore \log_7\beta = \sqrt{2} $ or $\log_7\beta = – \sqrt{2}$ (rejected).
Therefore, the common difference
$\begin{array}{cl}
= & \log_7\beta -\log_\beta \alpha \\
= & \log_7 \beta -\dfrac{\log_7 \alpha}{\log_7 \beta} \\
= & \log_7\beta -\dfrac{2-\log_7\beta}{\log_7\beta} \\
= & \sqrt{2} -\dfrac{2-\sqrt{2}}{\sqrt{2}} \\
= & 1
\end{array}$
2023-I-18
Ans: (a) $\log_7 \alpha = 2-\log_7 \beta$ (b) $1$
