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Q.
The value of $9^{1 / 3} \times 9^{1 / 9} \times 9^{1 / 27} \times ..........\infty$ is:
Sequences and Series
Solution:
$9^{\frac{1}{3}} \times 9^{\frac{1}{9}} \times 9^{\frac{1}{27}} \times \ldots . \infty=9^{\frac{1}{3}+\frac{1}{9}+\frac{1}{27}} \ldots \ldots \infty$
The powers of 9 form a GP with common
ratio $\frac{1}{3}$ and we know, sum of G.P. upto $\infty=\frac{a}{1-r}$
where ' $a$ ' is the first term and 'r' is the common ratio.
$9^{\frac{1}{3}} \times 9^{\frac{1}{9}} \times 9^{\frac{1}{27}} \times \ldots . . \infty=9^{\frac{1 / 3}{1-1 / 3}}=9^{\frac{1}{2}}=3$