Arithmetico geometric series

$\dfrac{2 + 6}{4^{100}} + \dfrac{2 + 2*6}{4^{99}} + \dfrac{2 + 3*6}{4^{98}} + \cdots + \dfrac{2 + 100*6}{4}$ $S=\dfrac{2 + 6}{4^{100}} + \dfrac{2 + 2*6}{4^{99}} + \dfrac{2 + 3*6}{4^{98}} + \cdots + \dfrac{2 + 100*6}{4}$ $S=2*(\dfrac{1}{4^{100}} + \dfrac{1}{4^{99}} + \dfrac{1}{4^{98}} + \cdots + \dfrac{1}{4})+6*(\dfrac{1}{4^{100}} + \dfrac{2 }{4^{99}} + \dfrac{3}{4^{98}} + \cdots + \dfrac{100}{4})$

$S=2X+6Y$

Now, $X$ is a simple $G.P.$ and $Y$ is an $A.G.P.$ .

$Y=\dfrac{1}{4^{100}} + \dfrac{2 }{4^{99}} + \dfrac{3}{4^{98}} + \cdots + \dfrac{100}{4}$

$4Y=\dfrac{1}{4^{99}} + \dfrac{2 }{4^{98}} + \dfrac{3}{4^{97}} + \cdots + 100$

Subtracting these two after shifting one place in $4Y$ we get,

$-3Y=\dfrac{1}{4^{100}} + \dfrac{1}{4^{99}} + \dfrac{1}{4^{98}} + \cdots + \dfrac{1}{4}-100$

$\implies -3Y=X-100$

$\implies Y=\dfrac{100-X}{3}$

$S=2X+6Y=2X+200-2X=200$