Let's analyze each summation to determine which one is equivalent to the given summation:
[tex]\[
\sum_{n=0}^3\left(-\frac{1}{2}\right)^n
\][/tex]
First, let's compute the value of the given summation:
[tex]\[
\left(-\frac{1}{2}\right)^0 + \left(-\frac{1}{2}\right)^1 + \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right)^3
\][/tex]
[tex]\[
= 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8}
\][/tex]
[tex]\[
= 1 - 0.5 + 0.25 - 0.125
\][/tex]
[tex]\[
= 0.625
\][/tex]
The sum is [tex]\(0.625\)[/tex]. Now let's compute the value of each option to check which one is equivalent.
### Option 1:
[tex]\[
\sum_{n=0}^3(-2)^{2n}
\][/tex]
Evaluating each term:
[tex]\[
(-2)^0 + (-2)^2 + (-2)^4 + (-2)^6
\][/tex]
[tex]\[
= 1 + 4 + 16 + 64
\][/tex]
[tex]\[
= 85
\][/tex]
### Option 2:
[tex]\[
\sum_{n=-4}^{-1}\left(\frac{1}{4}\right)(-2)^{-n}
\][/tex]
Evaluating each term:
[tex]\[
\left(\frac{1}{4}\right)(-2)^4 + \left(\frac{1}{4}\right)(-2)^3 + \left(\frac{1}{4}\right)(-2)^2 + \left(\frac{1}{4}\right)(-2)^1
\][/tex]
[tex]\[
= \left(\frac{1}{4}\right)(16) + \left(\frac{1}{4}\right)(-8) + \left(\frac{1}{4}\right)(4) + \left(\frac{1}{4}\right)(-2)
\][/tex]
[tex]\[
= 4 - 2 + 1 - 0.5
\][/tex]
[tex]\[
= 2.5
\][/tex]
### Option 3:
[tex]\[
\sum_{n=2}^5(-2)^n
\][/tex]
Evaluating each term:
[tex]\[
(-2)^2 + (-2)^3 + (-2)^4 + (-2)^5
\][/tex]
[tex]\[
= 4 - 8 + 16 - 32
\][/tex]
[tex]\[
= -20
\][/tex]
### Option 4:
[tex]\[
\sum_{n=0}^1(2)^{-n}
\][/tex]
Evaluating each term:
[tex]\[
(2)^0 + (2)^{-1}
\][/tex]
[tex]\[
= 1 + \frac{1}{2}
\][/tex]
[tex]\[
= 1.5
\][/tex]
Comparing the value of each option with the given summation:
- Given summation: [tex]\(0.625\)[/tex]
- Option 1: [tex]\(85\)[/tex]
- Option 2: [tex]\(2.5\)[/tex]
- Option 3: [tex]\(-20\)[/tex]
- Option 4: [tex]\(1.5\)[/tex]
None of these options yield the same result as the given summation. Therefore, none of the provided options is equivalent to:
[tex]\[
\sum_{n=0}^3\left(-\frac{1}{2}\right)^n
\][/tex]
