Question 23 and 24, Exercise 4.7
Solutions of Question 23 and 24 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 23
Sum to $n$ terms of the series (arithmetico-geometric series): $$1+2 \times 2+3 \times 2^{2}+4 \times 2^{3}+\ldots.$$
Solution.
Given arithmetic-geometric series: $$1+2 \times 2+3 \times 2^{2}+4 \times 2^{3}+\ldots$$
It can be written as $$ 1\times 1+2 \times 2+3 \times 2^{2}+4 \times 2^{3}+\ldots $$
The numbers $1,2,3,4,\ldots$ are in A.P. with $a=1$ and $d=1$.
The numbers $1, 2, 2^2, 2^3, \ldots$ are in G.P. with first term as 1 and $r=\frac{2}{1}=2$.
The sume of first $n$ terms of the arithmetico-geometric series is given by
\begin{align*}
S_{n}=\frac{a}{1-r}+d r \frac{\left(1-r^{n}\right)}{(1-r)^{2}}-\frac{(a+n d) r^{n}}{1-r}
\end{align*}
Thus
\begin{align*}
S_{n}&=\frac{1}{1-2}+(1)(2) \frac{\left(1-2^n\right)}{(1-2)^2}-\frac{(1+n (1)) (2)^{n}}{1-2} \\
&=-1+2-2\cdot 2^n+2^n+n\cdot 2^n \\
&= n\cdot 2^n -2^n + 1 \\
&=2^n(n-1)+1.
\end{align*}
This is the required sum. 
Question 24
Sum to $n$ terms of the series (arithmetico-geometric series): $$1+4 y+7 y^{2}+10 y^{3}+\ldots$$
Solution.
The given arithmetic-geometric series is: \[ 1 + 4y + 7y^2 + 10y^3 + \ldots \]
The numbers \(1, 4, 7, 10, \ldots\) are in A.P. with \(a = 1\) and \(d = 3\).
The numbers \(1, y, y^2, y^3, \ldots\) are in G.P. with first term \(1\) and \(r = y\).
The sum of the first \(n\) terms of the arithmetico-geometric series is given by: \[ S_n = \frac{a}{1 - r} + d r \frac{1 - r^n}{(1 - r)^2} - \frac{(a + n d) r^n}{1 - r} \]
This gives \begin{align*} S_n & = \frac{1}{1 - y} + 3 \cdot y \cdot \frac{1 - y^n}{(1 - y)^2} - \frac{(1 + n \cdot 3) y^n}{1 - y} \\ &= \frac{1}{1 - y} + \frac{3y(1 - y^n)}{(1 - y)^2} - \frac{(1 + 3n)y^n}{1 - y}\\ &= \frac{(1 - y)}{(1 - y)^2} + \frac{3y - 3y^{n+1}}{(1 - y)^2} - \frac{(y^n + 3ny^n)(1 - y)}{(1 - y)^2}\\ &= \frac{(1 - y)+(3y - 3y^{n+1})-(y^n + 3ny^n-y^{n+1} -3n y^{n+1})}{(1 - y)^2}\\ &= \frac{1 - y+3y - 3y^{n+1}-y^n - 3ny^n+y^{n+1} +3n y^{n+1}}{(1 - y)^2}\\ &= \frac{3n y^{n+1} - 2y^{n+1} - 3ny^n-y^n+2y+1 }{(1 - y)^2} \end{align*}
This is the required sum.
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