CAT 2003 Slot 2 — QA Question 46

$\begin{array}{rl}& \log m+\log \left(\frac{m^2}{n}\right)+\log \left(\frac{m^3}{n^2}\right)+\log \left(\frac{m^4}{n^3}\right)+\dots +\log \left(\frac{m^n}{n^{n-1}}\right)\\ & =\log \left[\frac{m\times m^2\times m^3\times \dots \times m^n}{1\times n\times n^2\times \dots \times n^{n-1}}\right]\\ & =\log \left[\frac{m^{1+2+3+4+\dots +n}}{n^{0+1+2+3+\dots +(n-1)}}\right]\\ & =\log \left[\frac{m^{\frac{n}{2}(n+1)}}{n^{\frac{n}{2}(n-1)}}\right]\\ & =\log {\left[\frac{m^{(n+1)}}{n^{(n-1)}}\right]}^{\frac{n}{2}}\end{array}$

Hence, option (d).