Solutions - 0150-04

(a)

$$0.2222\dots \gg \gg \frac{2}{10}+\frac{2}{100}+\frac{2}{1000}+\cdots$$

First term: $a_0=2/10$.

Common ratio: $r=1/10$

Geometric series summation formula:

$$S=\frac{a_0}{1-r}\gg \gg \frac{2/10}{1-1/10}\gg \gg \frac{2}{9}$$

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(b)

$$\begin{array}{c}0.49999\dots \gg \gg \frac{4}{10}+\frac{9}{100}+\frac{9}{1000}+\cdots \\ \\ \gg \gg \frac{4}{10}+(\frac{9}{100}+\frac{9}{1000}+\cdots )\end{array}$$

This is geometric starting with $9/100$.

First term: $a_0=9/100$.

Common ratio: $r=1/10$.

Geometric series summation formula:

$$S=\frac{9/100}{1-1/10}\gg \gg \frac{1}{10}$$

Add back the first term:

$$\gg \gg \frac{4}{10}+\frac{1}{10}\gg \gg \frac{1}{2}$$