A repeating decimal can always be expressed as a fraction....

A repeating decimal can always be expressed as a fraction. This problem shows how writing a repeating decimal as a geometric series enables you to find the fraction. (a) Write the repeating decimal $0.232323 \ldots$ as a geometric series using the fact that $0.232323 \ldots=$ $0.23+0.0023+0.000023+\cdots.$ (b) Use the formula for the sum of a geometric series to show that $0.232323 \ldots=23 / 99.$

A repeating decimal can always be expressed as a fraction. This problem shows how writing a repeating decimal as a geometric series enables you to find the fraction.
(a) Write the repeating decimal $0.232323 \ldots$ as a geometric series using the fact that $0.232323 \ldots=$ $0.23+0.0023+0.000023+\cdots.$
(b) Use the formula for the sum of a geometric series to show that $0.232323 \ldots=23 / 99.$

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Key Concepts

Repeating Decimal

A repeating decimal is a decimal number in which a digit or group of digits repeats infinitely. This concept is important because every repeating decimal can be expressed as a fraction, illustrating the connection between periodic decimal expansions and rational numbers.

Geometric Series

A geometric series is a sum of a sequence of terms where each term after the first is found by multiplying the previous one by a constant called the common ratio. In the context of repeating decimals, the infinite series representing the decimal has a clear structure with this constant ratio.

Sum of an Infinite Geometric Series

For a geometric series with a first term 'a' and a common ratio 'r' such that |r| < 1, the sum can be calculated using the formula S = a/(1 - r). This formula is crucial for converting a repeating decimal into a fraction by summing its geometric series representation.

Conversion of Repeating Decimals to Fractions

Expressing a repeating decimal as a geometric series allows the application of the sum formula for infinite geometric series. By identifying the repeating block as the first term and determining the appropriate common ratio, the decimal can be summed to yield its exact fractional form.

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