What is $0.9 + 0.09 + 0.009 + . . .$ equal to?

1

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1.01

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1.001

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1.1

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Solution

The correct option is A $1$
This can be written as
$\frac{9}{10} + \frac{9}{100} + \frac{9}{1000} . . . .$
$9 ((\frac{1}{10} + \frac{1}{100} + \frac{1}{1000} . .)$ . The bracket term forms an infinite G.P with first term $0.1$ and ratio $0.1$ .
So required sum
$= 9 \times \frac{0.1}{1 - 0.1} = \frac{0.9}{0.9} = 1$
Hence, option A is correct.

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