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Find the sum of the geometric sequence [tex]\(-3, 15, -75, 375, \ldots\)[/tex] when there are 7 terms. Select the correct answer below.

A. [tex]\(-39,063\)[/tex]
B. 39,063
C. 58,593
D. [tex]\(-58,593\)[/tex]


Sagot :

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]
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