Question

For the given sequence $\log a,\log \left(ab\right),\log \left(a{b}^2\right)$, where $a,b,c>0$, the ${10}^{th}$ term is

• A. $10\log \left(ab\right)$
• B. $\log \left(a{b}^{10}\right)$
• C. $9\log \left(ab\right)$
• D. $\log \left(a{b}^9\right)$

Answer 1

The correct option is D $\log \left(a{b}^9\right)$

Given sequence is $\log a,\log \left(ab\right),\log \left(a{b}^2\right)$
By observation,
$\log a+\log \left(a{b}^2\right)=\log \left(a^2{b}^2\right)=2\log \left(ab\right)$
Therefore, it is an A.P. with $A=\log a,D=\log \left(ab\right)-\log a=\log b$

Now,
$T_{10}=\log a+9\log b\Rightarrow T_{10}=\log a+\log b^9\therefore T_{10}=\log \left(a{b}^9\right)$