A) 9
B) 1
C) 3
D) none of these
Correct Answer: C
Solution :
\[{{9}^{1/3}}\times {{9}^{1/3}}\times {{9}^{1/27}}\times ....\infty \] \[={{(9)}^{\left[ \frac{1}{3}+\frac{1}{9}+\frac{1}{27}+....\infty \right]}}\] \[={{(9)}^{y}}\]where\[y=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+....\infty \] \[y=\frac{1}{3}\left[ 1+\frac{1}{3}+\frac{1}{{{3}^{2}}}+...\infty \right]\] \[=\frac{1}{3}\left[ \frac{1}{1-\frac{1}{3}} \right]\] [\[\because \]Sum of GP to\[\infty \]having first term a and ratio r is\[\frac{a}{1-r}\]] \[=\frac{1}{3}.\frac{3}{2}=\frac{1}{2}\] \[\therefore \]Sum\[={{(9)}^{1/2}}=3\]
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