Question 27 and 28, Exercise 4.7
Solutions of Question 27 and 28 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 27
Find sum to infinity of the series: $$5+\frac{7}{3}+\frac{9}{9}+\frac{11}{27}+\ldots$$.
Solution.
Given arithmetic-geometric series is: $$5+\frac{7}{3}+\frac{9}{9}+\frac{11}{27}+\ldots$$ It can be written as:
$$ 5\times 1+7\times\frac{1}{3}+9\times\frac{1}{9}+11\times\frac{1}{27}+\ldots $$
The numbers $5,7,9,11,4,\ldots$ are in A.P. with $a=5$ and $d=7-5=2$.
The numbers $1, \dfrac{1}{3}, \dfrac{1}{9}, \dfrac{1}{27}, \ldots$ are in G.P. with first term as 1 and $r=\dfrac{1/3}{1}=\dfrac{1}{3}$.
The sum of infinite arithmetico-geometric series is given by
$$
S_{\infty}=\frac{a}{1-r}+\frac{d r}{(1-r)^{2}}
$$
Thus
\begin{align*}
S_{\infty}&=\frac{5}{1-1/3}+ \frac{\left(2 \times 1/3\right)}{(1-1/3)^2} \\
&=\frac{5}{2/3}+ \frac{\left(2/3\right)}{(2/3)^2} \\
&=\frac{15}{2}+ \frac{3}{2} \\
&= 9
\end{align*}
This is the required sum. 
Question 28
Find sum to infinity of the series: $$1+\frac{2}{5}+\frac{3}{25}+\frac{4}{125}+\ldots$$
Solution.
The given arithmetic-geometric series is: \[ 1 + \frac{2}{5} + \frac{3}{25} + \frac{4}{125} + \ldots \]
It can be rewritten as:
\[
1 \times 1 + 2 \times \frac{1}{5} + 3 \times \frac{1}{25} + 4 \times \frac{1}{125} + \ldots
\]
The numbers \(1, 2, 3, 4, \ldots\) are in AP with \(a = 1\) and \(d = 1\).
The numbers \(1, \frac{1}{5}, \frac{1}{25}, \frac{1}{125}, \ldots\) are in GP with first term \(1\) and \(r = \frac{1/5}{1} = \frac{1}{5}\).
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 - \frac{1}{5}} + \frac{1 \times \frac{1}{5}}{(1 - \frac{1}{5})^{2}} \\ &= \frac{1}{4/5} + \frac{1/5}{\left(4/5\right)^{2}} \\ &= \frac{5}{4} + \frac{5}{16} \\ &= \frac{25}{16} \end{align*}
This is the required sum.
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