Question

The value of $9^{1/3}\times 9^{1/9}\times 9^{1/27}\times ..........\infty$ is:

• A. 9
• B. 1
• C. 3
• D. None of these

Answer 1

Answer: (C)

$9^{\frac{1}{3}}\times 9^{\frac{1}{9}}\times 9^{\frac{1}{27}}\times \dots .\infty =9^{\frac{1}{3}+\frac{1}{9}+\frac{1}{27}}\dots \dots \infty$
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.
$9^{\frac{1}{3}}\times 9^{\frac{1}{9}}\times 9^{\frac{1}{27}}\times \dots ..\infty =9^{\frac{1/3}{1-1/3}}=9^{\frac{1}{2}}=3$