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CAT 2003 QA - Retake

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Question:

What is the sum of n terms in the series

$\log m + \log \frac{m^{2}}{n} + \log \frac{m^{3}}{n^{2}} + \log \frac{m^{4}}{n^{3}} + \dots + \log \frac{m^{n}}{n^{n - 1}} ?$

Explanation:

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

Hence, option (d).

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