Question

Evaluate:


`\bf{\sum^(10)_(n-1)\:8(\cfrac{1}{4})^(n-1)`


`\bf{\overline{\underline{\overline{\underline{Please\:show\;work!}}}}`

Answer 1

Let S be the given sum, so

`S = 8 + 8 \left(\frac14\right) + 8 \left(\frac14\right)^2 + \cdots + 8 \left(\frac14\right)^9`

`\displaystyle S = 8 \left(1 + \frac14 + \frac1{4^2} + \cdots + \frac1{4^9}\right)`

Multiply both sides by 1/4.

`\displaystyle \frac S4 = 8 \left(\frac14 + \frac1{4^2} + \frac1{4^3} + \cdots + \frac1{4^(10)}\right)`

Subtract this from S to eliminate all the but the first term in S and the last term in S/4 :

`\displaystyle S - \frac S4 = 8 \left(1 - \frac1{4^(10)}\right)`

Solve for S :

`\displaystyle \frac{3S}4 = 8 \left(1 - \frac1{4^(10)}\right)`

`\displaystyle S = \frac{32}3 \left(1 - \frac1{4^(10)}\right) = \boxed{(349,525)/(32,768)}`