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What is the sum of 'n' terms in the series: $\log m + \log \frac{m^2}{n} + \log \frac{m^3}{n^2} + \log \frac{m^4}{n^3} + \dots + \log \frac{m^n}{n^{n-1}}?$

  1. $\log \left[\frac{n^{n-1}}{m^{(n+1)}} \right]^{\frac{n}{2}}$
  2. $\log \left[\frac{m^m}{n^n} \right]^{\frac{n}{2}}$
  3. $\log \left[\frac{m^{(1-n)}}{n^{(1-m)}} \right]^{\frac{n}{2}}$
  4. $\log \left[\frac{m^{(n+1)}}{n^{(n-1)}} \right]^{\frac{n}{2}}$
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log m + log (m^2/n) + log (m^3/n^2)+log(m^4/n^3 )+......log(m^n/n^n-1)
  
=> log(m* m^2/n * m^3/n^2 * m^4/n^3 ... *m^n/n^n-1) 
  
=> log(m^(1+2+3+4+..+n)/n^(1+2+3+4+..+n-1) )
  
=> log[m^(n*(n+1)/2)/n^(n*(n-1)/2]

=>log[m^(n+1)/n^(n-1)]^n/2

so option should be D

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