Solution:
(i) 
![\begin{array}{l} =22+3\left(3^{11}-1\right) /(3-1)\left[\text { by using the formula, } a\left(1-r^{n}\right) /(1-r)\right] \\ =22+3\left(3^{11}-1\right) / 2 \\ =[44+3(177147-1)] / 2 \\ =[44+3(177146)] / 2 \\ =265741 \end{array}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bb91c2c7993690f572331337414c3015_l3.png "Rendered by QuickLaTeX.com")
(ii) 
Let’s consider,
Where, 
Using the formula,
The sum of GP for 
Let’s now consider
Where, 
Using the formula,
The sum of GP for
Therefore,
![\begin{array}{l} \sum_{k=1}^{n}\left(2^{k}+3^{k-1}\right)=\left(2+2^{2}+2^{3}+\ldots+2^{n}\right)+\left(3^{0}+3^{1}+3^{2}+\ldots+3^{n}\right) \\ =2\left(2^{n}-1\right)+\left(3^{n}-1\right) / 2 \\ =1 / 2\left[2^{n+2}+3^{n}-4-1\right] \\ =1 / 2\left[2^{n+2}+3^{n}-5\right] \end{array}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-860700dd8e7a3beb950307be3806a110_l3.png "Rendered by QuickLaTeX.com")
Evaluate the following:
(i)
(ii) 
Solution:
(i)
(ii)
Let’s consider,
Where,
Using the formula,
The sum of GP for
Let’s now consider
Where,
Using the formula,
The sum of GP for
Therefore,
