Change the decimal to a fraction 0.123123123…

Answer 1

Let's call the given decimal number x. To convert this decimal into a fraction, we can use the technique of infinite geometric series.

The decimal 0.123123123... can be expressed as the sum of an infinite geometric series with the first term 0.123 and a common ratio of 0.001:

x = 0.123 + 0.001(0.123) + 0.001^2(0.123) + 0.001^3(0.123) + ...

To find the sum of this infinite geometric series, we can use the formula for the sum of an infinite geometric series:

S = a / (1 - r)

Where S is the sum of the series, a is the first term, and r is the common ratio.

In this case, a = 0.123 and r = 0.001:

S = 0.123 / (1 - 0.001)
S = 0.123 / 0.999
S = 123/999

Therefore, the decimal 0.123123123... is equal to the fraction 123/999.