Let S be the given sum, so
[tex]S = 8 + 8 \left(\dfrac14\right) + 8 \left(\dfrac14\right)^2 + \cdots + 8 \left(\dfrac14\right)^9[/tex]
[tex]\displaystyle S = 8 \left(1 + \dfrac14 + \frac1{4^2} + \cdots + \frac1{4^9}\right)[/tex]
Multiply both sides by 1/4.
[tex]\displaystyle \frac S4 = 8 \left(\frac14 + \frac1{4^2} + \frac1{4^3} + \cdots + \frac1{4^{10}}\right)[/tex]
Subtract this from S to eliminate all the but the first term in S and the last term in S/4 :
[tex]\displaystyle S - \frac S4 = 8 \left(1 - \frac1{4^{10}}\right)[/tex]
Solve for S :
[tex]\displaystyle \frac{3S}4 = 8 \left(1 - \frac1{4^{10}}\right)[/tex]
[tex]\displaystyle S = \frac{32}3 \left(1 - \frac1{4^{10}}\right) = \boxed{\frac{349,525}{32,768}}[/tex]
