A) 6
B) 36
C) 216
D) \[\infty \]
Correct Answer: C
Solution :
[c] \[X={{6}^{\left( \frac{1}{2}+\frac{1}{2}+\frac{3}{8}+\frac{1}{4}+..... \right)}}\] \[={{6}^{\left[ \left( 1\times \frac{1}{2} \right)+\left( 2\times \frac{1}{4} \right)+\left( 3\times \frac{1}{8} \right)+\left( 4\times \frac{1}{16} \right)+...... \right]}}\] \[\because \] It is arithmetic-geometric progression, \[\therefore a=\frac{1}{2};\,\,\,d=1\] & \[r=\frac{1}{2}\] \[\Rightarrow \,\,\,\,\,\,\,\,X={{6}^{\left[ \frac{a}{1-r}+\frac{dr}{{{(1-r)}^{2}}} \right]}}={{6}^{\left[ \frac{\frac{1}{2}}{1-\frac{1}{2}}+\frac{1\times \frac{1}{2}}{{{(1-\frac{1}{2})}^{2}}} \right]}}={{6}^{3}}=216\]
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