To find the sum of the first 7 terms in the geometric series [tex]\((-1), 2, (-4), 8, \ldots\)[/tex], we need to use the formula for the sum of the first [tex]\( n \)[/tex] terms of a geometric series. Let's break this process down step-by-step:
1. Identify the first term [tex]\( a_1 \)[/tex] and the common ratio [tex]\( r \)[/tex]:
- The first term [tex]\( a_1 \)[/tex] of the series is [tex]\(-1\)[/tex].
- To find the common ratio [tex]\( r \)[/tex], we divide the second term by the first term. That is [tex]\( r = \frac{2}{-1} = -2 \)[/tex].
2. Determine the number of terms [tex]\( n \)[/tex]:
- Here, we are asked to find the sum of the first 7 terms, so [tex]\( n = 7 \)[/tex].
3. Use the formula for the sum of the first [tex]\( n \)[/tex] terms of a geometric series:
[tex]\[
S_n = a_1 \frac{1 - r^n}{1 - r}
\][/tex]
4. Substitute the values into the formula:
[tex]\[
S_7 = -1 \cdot \frac{1 - (-2)^7}{1 - (-2)}
\][/tex]
Simplifying the denominator:
[tex]\[
S_7 = -1 \cdot \frac{1 - (-2)^7}{1 + 2}
\][/tex]
[tex]\[
S_7 = -1 \cdot \frac{1 - (-128)}{3}
\][/tex]
5. Calculate [tex]\( (-2)^7 \)[/tex]:
- Since [tex]\( -2 \)[/tex] raised to the 7th power equals [tex]\(-128\)[/tex], the equation becomes:
[tex]\[
S_7 = -1 \cdot \frac{1 - (-128)}{3}
\][/tex]
6. Simplify the expression further:
[tex]\[
S_7 = -1 \cdot \frac{1 + 128}{3}
\][/tex]
[tex]\[
S_7 = -1 \cdot \frac{129}{3}
\][/tex]
[tex]\[
S_7 = -1 \cdot 43
\][/tex]
[tex]\[
S_7 = -43
\][/tex]
Therefore, the sum of the first 7 terms of the series is [tex]\(-43\)[/tex].
The correct answer is A. -43.
