Question

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

• A. a GP
• B. an AP
• C. a HP
• D. Both a GP and a HP

Answer 1

The correct option is B

an AP

Explanation of the correct option

Arithmetic Progression (AP) is a sequence of numbers in order, in which the difference $\left(d\right)$ between any two consecutive numbers is a constant value.

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

$$\begin{gathered} d=T_2-T_1 \\ =\log \left(\frac{a^2}{b}\right)-\log \left(a\right) \\ =2\log a-\log b-\log a \\ =\log a-\log b \\ =\log \left(\frac{a}{b}\right) \\ d=T_3-T_2 \\ =\log \left(\frac{a^3}{b^2}\right)-\log \left(\frac{a^2}{b}\right) \\ =3\log a-2\log b-2\log a+\log b \\ =\log a-\log b \\ =\log \left(\frac{a}{b}\right) \end{gathered}$$

Hence the difference $\left(d\right)$ between consecutive terms is same.

Therefore given sequence is in AP.

Hence, the correct option is option (A).