====== 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. GOOD m(







=====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. GOOD









====Go to ====

<text align="left"><btn type="primary">[[math-11-nbf:sol:unit04:ex4-7-p13|< Question 25 & 26]]</btn></text>
<text align="right"><btn type="success">[[math-11-nbf:sol:unit04:ex4-7-p15|Question 29 & 30 >]]</btn></text>