# Question 29 and 30, Exercise 4.7

Solutions of Question 29 and 30 of Exercise 4.7 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 29

Find sum to infinity of the series: $$1+4 x+7 x^{2}+10 x^{3}+\ldots$$

** Solution. **

The given arithmetic-geometric series is:

\[
1 + 4x + 7x^2 + 10x^3 + \ldots
\]

It can be rewritten as:

\[
1 \times 1 + 4 \times x + 7 \times x^2 + 10 \times x^3 + \ldots
\]

The numbers \(1, 4, 7, 10, \ldots\) are in AP with \(a = 1\) and \(d = 4 - 1 = 3\).

The numbers \(1, x, x^2, x^3, \ldots\) are in GP with first term \(1\) and \(r = x\).

The sum of the infinite arithmetico-geometric series is given by:

\[
S_{\infty} = \frac{a}{1 - r} + \frac{d r}{(1 - r)^{2}}
\]

Thus, we have:

\begin{align*}
S_{\infty} &= \frac{1}{1 - x} + \frac{(3 \times x)}{(1 - x)^{2}} \\
&= \frac{1}{1 - x} + \frac{3x}{(1 - x)^2}\\
&= \frac{1-x+3x}{(1 - x)^2}\\
&= \frac{1+2x}{(1 - x)^2}
\end{align*}

This is the required sum.

## Question 30

Find sum to infinity of the series: $$3+\frac{6}{10}+\frac{9}{100}+\frac{12}{1000}+\ldots$$

** Solution. **

The given arithmetic-geometric series is:

\[
3 + \frac{6}{10} + \frac{9}{100} + \frac{12}{1000} + \ldots
\]

It can be rewritten as:

\[
3 \times 1 + 6 \times \frac{1}{10} + 9 \times \frac{1}{100} + 12 \times \frac{1}{1000} + \ldots
\]

\(3, 6, 9, 12, \ldots\) are in AP with \(a = 3\) and \(d = 6 - 3 = 3\).

\(1, \frac{1}{10}, \frac{1}{100}, \frac{1}{1000}, \ldots\) are in GP with first term \(1\) and \(r = \frac{1}{10}\).

The sum of the infinite arithmetico-geometric series is given by:

\[
S_{\infty} = \frac{a}{1 - r} + \frac{d r}{(1 - r)^{2}}
\]

Thus, we have:

\begin{align*}
S_{\infty} &= \frac{3}{1 - \frac{1}{10}} + \frac{(3 \times \frac{1}{10})}{\left(1 - \frac{1}{10}\right)^{2}} \\
&= \frac{3}{9/10} + \frac{3/10}{\left(9/10\right)^{2}} \\
&= \frac{10}{3} + \frac{10}{27} \\
&= \frac{100}{27}
\end{align*}

This is the required sum.

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