# 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.

## Question 14(i)

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 $S_{\infty} =\dfrac{4}{9}$.

## Question 14(ii)

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 $.

## Question 14(iii)

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, $S_{\infty}= \frac{5}{9}.$

## Question 14(iv)

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, $S_{\infty}= \frac{2}{3}.$

## Question 14(v)

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, $S_{\infty}= \frac{5}{33}.$

## Question 14(vi)

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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