Infinite geometric interpretation

Converting recurring decimals to fractions is a classic application of infinite geometric series, a key skill in the HSC Extension 1 Sequences and Series topic. By splitting the decimal into a terminating part and a recurring part, we can use the sum-to-infinity formula for a geometric progression to express the number exactly as a fraction. This problem demonstrates that technique for a decimal with a delayed repeating block.

Problem Statement

Evaluate

$$0.4\overline{2}=0.42222\ldots$$

as a fraction.

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Hints

Write as $0.4+0.02222\ldots$ and convert the recurring part using a GP.

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Solutions

Following the hint, we write $0.4\overline{2}$ as $0.4 + 0.02222\ldots$. The recurring decimal $0.02222\ldots$ can be seen as the infinite geometric series $0.02 + 0.002 + 0.0002 + \dots$ with first term $a = 0.02$ and common ratio $r = 0.1$. Since $|r| = 0.1 < 1$, the series converges, and applying the sum-to-infinity formula $S = \frac{a}{1-r}$ gives

$$0.02222\ldots=\frac{2}{90}=\frac{1}{45}.$$

Now we combine this with the non-repeating part, $0.4 = \frac{2}{5}$, and add the two fractions:

$$0.4\overline{2}=\frac{2}{5}+\frac{1}{45}=\frac{18}{45}+\frac{1}{45}=\frac{19}{45}.$$

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Takeaways

• Isolate the repeating block as a geometric series; its first term $a$ and common ratio $r = 10^{-k}$ (where $k$ is the length of the repeating block) let you apply the sum-to-infinity formula.
• The sum-to-infinity formula $S = \frac{a}{1-r}$ (valid for $|r|<1$) converts the recurring part directly to a fraction.
• Adding the terminating decimal part, also expressed as a fraction, gives the final rational representation.

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Further Readings

HSC Trigonometry, HSC Last Resorts, HSC Polys Ext 1, HSC Complex Numbers