Give two rational numbers lying between

Question:

Give two rational numbers lying between 0.515115111511115...0.5353353335...

Solution:

Let $a=0.515115111511115 \ldots$ and $b=0.535335333533335 \ldots$

Here the decimal representation of a and are non-terminating and non-repeating. So we observe that in first decimal place a and b have the same digit  but digit in the second place of their decimal representation are distinct. And the number a has 1 and b has 3. So a < b.

Hence two rational numbers are $0.5152,0.532$ lying between $0.515115111511115 \ldots$ and $0.535335333533335 \ldots$

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