Question 14, Exercise 4.5

Solutions of Question 14 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Find fractional notation for the infinite geometric series; $0.444...$

Solution.

We can express the decimal as $$0.444... = 0.4+0.04+0.004+...$$ This is infinite geometric series with $a_1=0.4$, $r=\frac{0.04}{0.4}=0.1$.
Since $|r|=0.1 < 1$, this series has the sum: \begin{align*} S-\infty & = \frac{a_1}{1-r} \\ & = \frac{0.4}{1.0.1} = \frac{0.4}{0.9} \\ & = \frac{4}{9}. \end{align*} Hence $S_{\infty} =\dfrac{4}{9}$.

Find fractional notation for the infinite geometric series; $9.99999 ...$

Solution.

We can express the decimal as $$0.99999 ... = 0.9+0.09+0.009+...$$ This is infinite geometric series with $a_1=0.9$, $r=\frac{0.09}{0.9}=0.1$.
Since $|r|=0.1 < 1$, this series has the sum: \begin{align*} S-\infty & = \frac{a_1}{1-r} \\ & = \frac{0.9}{1.0.1} = \frac{0.9}{0.9} \\ & = 1 \end{align*} Hence $S_{\infty}= 1 $.

Find fractional notation for the infinite geometric series; $0.5555 \ldots$

Solution.

We can express the decimal as $$0.5555 \ldots = 0.5 + 0.05 + 0.005 + \ldots$$ This is an infinite geometric series with $a_1 = 0.5$ and $r = \frac{0.05}{0.5} = 0.1$.
Since $|r| = 0.1 < 1$, this series has the sum: \begin{align*} S_\infty &= \frac{a_1}{1 - r} \\ &= \frac{0.5}{1 - 0.1} \\ &= \frac{0.5}{0.9} \\ &= \frac{5}{9}. \end{align*} Hence, $S_{\infty}= \frac{5}{9}.$

Find fractional notation for the infinite geometric series; $0.6666 \ldots$

Solution.

We can express the decimal as $$0.6666 \ldots = 0.6 + 0.06 + 0.006 + \ldots$$ This is an infinite geometric series with $a_1 = 0.6$ and $r = \frac{0.06}{0.6} = 0.1$
Since $|r| = 0.1 < 1$, this series has the sum: \begin{align*} S_\infty &= \frac{a_1}{1 - r} \\ &= \frac{0.6}{1 - 0.1} \\ &= \frac{0.6}{0.9} \\ &= \frac{6}{9} \\ &= \frac{2}{3}. \end{align*} Hence, $S_{\infty}= \frac{2}{3}.$

Find fractional notation for the infinite geometric series; $0.15151515 \ldots$

Solution.

We can express the decimal as $$0.151515 \ldots = 0.15 + 0.0015 + 0.000015 + \ldots$$ This is an infinite geometric series with $a_1 = 0.15$ and $r = \frac{0.0015}{0.15} = 0.01$
Since $|r| = 0.01 < 1$, this series has the sum: \begin{align*} S_\infty &= \frac{a_1}{1 - r} \\ &= \frac{0.15}{1 - 0.01} \\ &= \frac{0.15}{0.99} \\ &= \frac{15}{99} \\ &= \frac{5}{33}. \end{align*} Hence, $S_{\infty}= \frac{5}{33}.$

Find fractional notation for the infinite geometric series; $0.12121212 \ldots$

Solution.

We can express the decimal as $$0.121212 \ldots = 0.12 + 0.0012 + 0.000012 + \ldots$$ This is an infinite geometric series with $a_1 = 0.12$ and $r = \frac{0.0012}{0.12} = 0.01$
Since $|r| = 0.01 < 1$, this series has the sum: \begin{align*} S_\infty &= \frac{a_1}{1 - r} \\ &= \frac{0.12}{1 - 0.01} \\ &= \frac{0.12}{0.99} \\ &= \frac{12}{99} \\ &= \frac{4}{33}. \end{align*} Hence, $S_{\infty} = \frac{4}{33}$

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