Question

The sequence $\log a,\log \left(\frac{a^2}{b}\right),\log \left(\frac{a^3}{b^2}\right),....$ is

• A. GP
• B. AP
• C. HP
• D. GP and HP

Answer 1

The correct option is B

AP

Explanation for the correct option:

$\log a,\log \left(\frac{a^2}{b}\right),\log \left(\frac{a^3}{b^2}\right),....$

Let $x=\log a$

$$\begin{gathered} y=\log \left(\frac{a^2}{b}\right)=2\log a-\log b\mathbf{}\left[\because \log \left(\frac{m}{n}\right)=\mathrm{logm}-\mathrm{logn},{\mathrm{logm}}^a=\mathrm{alogm}\right] \\ z=\log \left(\frac{a^3}{b^2}\right)=3\log a-2\log b \end{gathered}$$

Suppose $x,y,z$ are in AP. Then sum of $x$ and $z$ terms is equal to $2y.$

$$\begin{gathered} x+z=\log a+3\log a-2\log b \\ =4\log a-2\log b \\ =2(2\log a-\log b) \\ =2y\mathbf{}\left[\because 2\mathrm{loga}-\mathrm{logb}=y\right] \end{gathered}$$

It satisfies the condition of AP.

Hence the correct option is option B.