$x y$ is a number that is divided by $a b$ where $x y < a b$ and gives a result $0 . x y x y x y x y \dots$ then find $a b$ .

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Solution

Solution.
The sequence $0 . x y x y x y x y \dots$ can be represented as:
$(\frac{x y}{100}) + (\frac{x y}{10000}) + (\frac{x y}{100000}) + \dots = \frac{x y}{100} \times (1 + \frac{1}{100} + \frac{1}{1000} + \dots .)$
Here, It is observed that the second part is a geometric series with the first term $a = 1$ and the common ratio of $r = \frac{1}{100}$ .
The sum of infinite geometric series is computed as,
$S_{n} = \frac{a}{1 - r}, r < 1 = \frac{x y}{100} \times (\frac{1}{1 - \frac{1}{100}}) = \frac{x y}{100} \times \frac{100}{99} = \frac{x y}{99}$
$∵ \frac{x y}{a b} = \frac{x y}{99} = 0 . x y x y x y x y . . . ∴ a b = 99$
Hence, the value of $a b$ is $99$ .

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