Question

Sum of n terms of the series $loga+log\left(\frac{a^2}{b}\right)+log\left(\frac{a^3}{b^2}\right)+log\left(\frac{a^4}{b^3}\right)+\cdots \infty$ is:

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

Answer 1

The correct option is B $log{\left\{\frac{a^{n+1}}{b^{n-1}}\right\}}^{n/2}$

$loga+log\frac{a^2}{b}+log\frac{a^3}{b^2}+log\frac{a^4}{b^3}+\cdots +log\frac{a^n}{b^{n-1}}$
$=log(a.\frac{a^2}{b}.\frac{a^3}{b^2}.\frac{a^4}{b^3}\cdots \frac{a^n}{n^{n-1}})$
$=log\left[\frac{a^{(1+2+3+4+\cdots +n)}}{b^{(1+2+3+4+\cdots +(n-1\left)\right)}}\right]$
$=log\left[\frac{a^{\frac{n(n+1)}{2}}}{b^{\frac{(n-1)n}{2}}}\right]=log{\left[\frac{a^{(n+1)}}{b^{(n-1)}}\right]}^{n/2}$