To find the sum of the geometric sequence [tex]\(-3, 15, -75, 375, \ldots\)[/tex] for 7 terms, we need to identify the first term, the common ratio, and then use the formula for the sum of the first [tex]\( n \)[/tex] terms of a geometric sequence.
1. Identify the first term ([tex]\( a \)[/tex]) and the common ratio ([tex]\( r \)[/tex]):
- The first term [tex]\( a \)[/tex] is [tex]\(-3\)[/tex].
- To find the common ratio [tex]\( r \)[/tex], we divide the second term by the first term: [tex]\( r = \frac{15}{-3} = -5 \)[/tex].
2. Write down the sum formula:
The sum [tex]\( S_n \)[/tex] of the first [tex]\( n \)[/tex] terms of a geometric sequence is given by:
[tex]\[
S_n = a \frac{1 - r^n}{1 - r}
\][/tex]
where [tex]\( a \)[/tex] is the first term, [tex]\( r \)[/tex] is the common ratio, and [tex]\( n \)[/tex] is the number of terms.
3. Substitute the identified values into the formula:
- [tex]\( a = -3 \)[/tex]
- [tex]\( r = -5 \)[/tex]
- [tex]\( n = 7 \)[/tex]
So, the sum of the first 7 terms is:
[tex]\[
S_7 = -3 \frac{1 - (-5)^7}{1 - (-5)}
\][/tex]
4. Calculate the expression step by step:
- First, calculate [tex]\( (-5)^7 \)[/tex]:
[tex]\[
(-5)^7 = -78125
\][/tex]
- Next, substitute [tex]\( (-5)^7 \)[/tex] back into the sum formula:
[tex]\[
S_7 = -3 \frac{1 - (-78125)}{1 + 5}
\][/tex]
- Simplify the numerator and denominator:
[tex]\[
S_7 = -3 \frac{1 + 78125}{6}
\][/tex]
[tex]\[
S_7 = -3 \frac{78126}{6}
\][/tex]
- Finally, perform the division and multiplication:
[tex]\[
\frac{78126}{6} = 13021
\][/tex]
[tex]\[
S_7 = -3 \times 13021 = -39063
\][/tex]
Thus, the sum of the geometric sequence when there are 7 terms is [tex]\(-39,063\)[/tex]. The correct answer is:
[tex]\[
\boxed{-39,063}
\][/tex]
