To identify which geometric series results in a sum of [tex]\(-69,905\)[/tex], let's analyze each option step by step.
### Option A
[tex]\[
\sum_{k=0}^{10} \frac{1}{5}(-4)^k
\][/tex]
This is a geometric series with the first term [tex]\( a = \frac{1}{5} \)[/tex] and the common ratio [tex]\( r = -4 \)[/tex]. The sum [tex]\( S \)[/tex] of the first [tex]\( n+1 \)[/tex] terms of a geometric series can be calculated using the formula:
[tex]\[
S = a \frac{1-r^{n+1}}{1-r}
\][/tex]
Here, [tex]\( n = 10 \)[/tex], [tex]\( a = \frac{1}{5} \)[/tex], and [tex]\( r = -4 \)[/tex]. Substituting these values:
[tex]\[
S = \frac{1}{5} \frac{1-(-4)^{11}}{1-(-4)}
\][/tex]
### Option B
[tex]\[
\sum_{k=0}^{11} -\frac{1}{4}(5)^k
\][/tex]
This series has [tex]\( a = -\frac{1}{4} \)[/tex] and [tex]\( r = 5 \)[/tex]. Using the geometric series sum formula:
[tex]\[
S = a \frac{1-r^{n+1}}{1-r}
\][/tex]
With [tex]\( n = 11 \)[/tex], [tex]\( a = -\frac{1}{4} \)[/tex], and [tex]\( r = 5 \)[/tex]:
[tex]\[
S = -\frac{1}{4} \frac{1-5^{12}}{1-5}
\][/tex]
### Option C
[tex]\[
\sum_{k=0}^7 \frac{1}{4}(-5)^k
\][/tex]
This series has [tex]\( a = \frac{1}{4} \)[/tex] and [tex]\( r = -5 \)[/tex]. Using the geometric series sum formula:
[tex]\[
S = a \frac{1-r^{n+1}}{1-r}
\][/tex]
With [tex]\( n = 7 \)[/tex], [tex]\( a = \frac{1}{4} \)[/tex], and [tex]\( r = -5 \)[/tex]:
[tex]\[
S = \frac{1}{4} \frac{1-(-5)^8}{1-(-5)}
\][/tex]
### Option D
[tex]\[
\sum_{k=0}^9 -\frac{1}{5}(4)^k
\][/tex]
This series has [tex]\( a = -\frac{1}{5} \)[/tex] and [tex]\( r = 4 \)[/tex]. Using the geometric series sum formula:
[tex]\[
S = a \frac{1-r^{n+1}}{1-r}
\][/tex]
With [tex]\( n = 9 \)[/tex], [tex]\( a = -\frac{1}{5} \)[/tex], and [tex]\( r = 4 \)[/tex]:
[tex]\[
S = -\frac{1}{5} \frac{1-4^{10}}{1-4}
\][/tex]
After analyzing the given options with their respective formulas, we obtain specific numerical sums. The series which results in the sum matching [tex]\(-69,905\)[/tex] is:
[tex]\[
\sum_{k=0}^9 -\frac{1}{5}(4)^k
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
