Question

The first term of an A.P is $\log a$ and the second term is $\log b$. Express the sum to $n$ terms as a logarithm.

• A. $\log \left(\frac{b^{n-1}}{a^{n-3}}\right)$
  
• B. $\log {\left(\frac{b^{n-1}}{a^{n-3}}\right)}^{n/2}$
  
• C. $\log {\left(\frac{b^{n-1}}{a^{n-3}}\right)}^n$
• D. $\log {\left(\frac{b^{n-3}}{a^{n-1}}\right)}^{n/2}$

Answer 1

Answer: (B)

$\log {\left(\frac{b^{n-1}}{a^{n-3}}\right)}^{n/2}$

Given that the first term is $\log a$ and second term is $\log b$

So the common difference is $\log b-\log a$

The sum of first $n$ terms is $n\times \log \left(a\right)+(\log b-\log a)(1+2+3+.....n-1)$

$\Rightarrow n\times \log \left(a\right)+\frac{n(n-1)}{2}(\log b-\log a)=\frac{1}{2}\left(n\right)(3-n)\log a+\frac{n(n-1)}{2}\log b=\log {\left(\frac{b^{n-1}}{a^{n-3}}\right)}^{\frac{n}{2}}$

Therefore option $B$ is correct