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If $ 1 \frac{1}{2} \times 2 \frac{2}{3} \times 3 \frac{3}{4} \times \ldots \times 199 \frac{199}{200}=\frac{1}{m}, $ then find the value of $ m $.
Given:
$1 \frac{1}{2} \times 2 \frac{2}{3} \times 3 \frac{3}{4} \times \ldots \times 199 \frac{199}{200}=\frac{1}{m}$
To do:
We have to find the value of $m$.
Solution:
$ \begin{array}{l}
1\frac{1}{2} \times 2\frac{2}{3} \times 3\frac{3}{4} \times ......\times 199\frac{199}{200} =\frac{1}{m}\\
\\
\frac{1\times 2+1}{2} \times \frac{2\times 3+2}{3} \times \frac{3\times 4+3}{4} \times ......\times \frac{199\times 200+199}{200} =\frac{1}{m}\\
\\
\frac{3}{2} \times \frac{8}{3} \times \frac{15}{4} \times ......\times \frac{199( 200+1)}{200} =\frac{1}{m}\\
\\
\frac{3}{2} \times \frac{8}{3} \times \frac{15}{4} \times ......\times \frac{199( 201)}{200} =\frac{1}{m}\\
\\
\frac{1\times 2\times 3\times ....\times 198\times 199\times 201}{2} =\frac{1}{m}\\
\\
\frac{199!( 201)}{2} =\frac{1}{m}\\
\\
m=\frac{2}{199!( 201)}
\end{array}$
The value of $m$ is $\frac{2}{199!(201)}$.
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