To evaluate the given geometric series [tex]\(\sum_{i=1}^8 2^{i-1}\)[/tex], let's start by understanding the series and using the formula for the sum of a geometric series.
A geometric series is a series of the form:
[tex]\[ a + ar + ar^2 + ar^3 + \cdots + ar^{n-1} \][/tex]
In this case, the series is:
[tex]\[ 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 \][/tex]
Here:
- The first term [tex]\( a \)[/tex] is [tex]\( 2^{1-1} = 2^0 = 1 \)[/tex].
- The common ratio [tex]\( r \)[/tex] is [tex]\( 2 \)[/tex].
- The number of terms [tex]\( n \)[/tex] is [tex]\( 8 \)[/tex].
We can use the formula for the sum of the first [tex]\( n \)[/tex] terms of a geometric series:
[tex]\[ S_n = a \frac{r^n - 1}{r - 1} \][/tex]
Plugging in the values:
- [tex]\( a = 1 \)[/tex]
- [tex]\( r = 2 \)[/tex]
- [tex]\( n = 8 \)[/tex]
We get:
[tex]\[ S_8 = 1 \frac{2^8 - 1}{2 - 1} \][/tex]
Calculate [tex]\( 2^8 \)[/tex]:
[tex]\[ 2^8 = 256 \][/tex]
Now, substitute back into the formula:
[tex]\[ S_8 = 1 \frac{256 - 1}{1} \][/tex]
[tex]\[ S_8 = 1 \cdot 255 \][/tex]
[tex]\[ S_8 = 255 \][/tex]
Therefore, the sum of the given geometric series [tex]\(\sum_{i=1}^8 2^{i-1}\)[/tex] is [tex]\( 255 \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{255} \][/tex]
