Question

The value of the infinite product $6^{\frac{1}{2}}\times 6^{\frac{1}{2}}\times 6^{\frac{3}{8}}\times 6^{\frac{1}{4}}\times .........$ is

• A. 6
• B. 36
• C. 216
• D. $\infty$

Answer 1

The correct option is B 36

$6^{\frac{1}{2}}\times 6^{\frac{1}{2}}\times 6^{\frac{3}{8}}\times 6^{\frac{1}{4}}\times ...............$

$=6^{\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+...................}$

$=6^{\frac{1}{2}(1+\frac{2}{2}+\frac{3}{2^2}+\frac{4}{2^3}+......................)}$

Let $S=1+\frac{2}{2}+\frac{3}{2^2}+\frac{4}{2^3}+............$, then

$\frac{S}{2}=S-\frac{S}{2}$

or, $\frac{S}{2}=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...........$

or, $\frac{S}{2}=\frac{1}{1-\frac{1}{2}}$ ..... [Using sum of infinite terms of an G.P]

or, $S=4$

Then we have,

$6^{\frac{1}{2}}\times 6^{\frac{1}{2}}\times 6^{\frac{3}{8}}\times 6^{\frac{1}{4}}\times ...............$

$=6^{\frac{1}{2}\times 4}=6^2=36$

Hence, B is the correct option.