Question

If $\underset{(i\ne j\ne k)}{\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }\sum _{k=0}^{\infty }\frac{1}{3^i{3}^j{3}^k}}=\frac{27\times A}{208}$; where $A$ is single digit natural number, then find the value of $A$

Answer 1

Answer: 3

No condition on $i,j,k$
$=\frac{3}{2}\times \frac{3}{2}\times \frac{3}{2}=\frac{27}{8}$
when any two of $i,j,k$ are equal
$=\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }\frac{1}{3^{2i}}\cdot \frac{1}{3^j}=\frac{3}{2}\left[1+\frac{1}{9}+\dots \dots .\right]=\frac{3}{2}\times \frac{9}{8}=\frac{27}{16}$
When all three are equal
$=1+\frac{1}{27}+\frac{1}{27^2}+\dots \dots \infty$
$=\frac{27}{26}$
$\text{ Required }=\frac{27}{8}-3\cdot \frac{27}{16}+\frac{2\cdot 27}{26}=\frac{81}{208}$