A) \[\frac{n}{2}\log \left( \frac{{{a}^{n}}}{{{b}^{n}}} \right)\]
B) \[\frac{n}{2}\log \left( \frac{{{a}^{n+1}}}{{{b}^{n}}} \right)\]
C) \[\frac{n}{2}\log \left( \frac{{{a}^{n+1}}}{{{b}^{n-1}}} \right)\]
D) \[\frac{n}{2}\log \left( \frac{{{a}^{n+1}}}{{{b}^{n+1}}} \right)\]
Correct Answer: C
Solution :
The given series is \[\log a+\log \left( \frac{{{a}^{2}}}{b} \right)+\log \left( \frac{{{a}^{3}}}{{{b}^{2}}} \right)+\log \left( \frac{{{a}^{4}}}{{{b}^{3}}} \right)+......+\log \left( \frac{{{a}^{n}}}{{{b}^{n-1}}} \right)\] This is an A.P. with first term and the common difference \[\log \left( \frac{{{a}^{2}}}{b} \right)-\log a=\log \left( \frac{a}{b} \right)\] Therefore the sum of terms is \[\frac{n}{2}\left[ \log a+\log \left( \frac{{{a}^{n}}}{{{b}^{n-1}}} \right) \right]=\frac{n}{2}\log \left( \frac{{{a}^{n+1}}}{{{b}^{n-1}}} \right)\]. Trick: Check for.You need to login to perform this action.
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