Answer :
To find the sum of the geometric sequence [tex]\(-4, 24, -144, \ldots\)[/tex] if there are 7 terms, we follow these steps:
1. Identify the first term and common ratio:
- The first term ([tex]\(a_1\)[/tex]) of the sequence is [tex]\(-4\)[/tex].
- The common ratio ([tex]\(r\)[/tex]) can be found by dividing the second term by the first term. So,
[tex]\[
r = \frac{24}{-4} = -6
\][/tex]
2. Use the formula for the sum of a geometric sequence:
The sum of the first [tex]\(n\)[/tex] terms of a geometric sequence is given by the formula:
[tex]\[
S_n = a_1 \frac{1 - r^n}{1 - r}
\][/tex]
where:
- [tex]\(S_n\)[/tex] is the sum of the first [tex]\(n\)[/tex] terms,
- [tex]\(a_1\)[/tex] is the first term,
- [tex]\(r\)[/tex] is the common ratio,
- [tex]\(n\)[/tex] is the number of terms.
3. Substitute the known values into the formula:
- Here, [tex]\(a_1 = -4\)[/tex], [tex]\(r = -6\)[/tex], and [tex]\(n = 7\)[/tex].
Plug these values into the formula:
[tex]\[
S_7 = -4 \frac{1 - (-6)^7}{1 - (-6)}
\][/tex]
4. Calculate [tex]\( (-6)^7 \)[/tex]:
[tex]\[
(-6)^7 = -279936
\][/tex]
5. Calculate the sum:
- Denominator: [tex]\(1 - (-6) = 1 + 6 = 7\)[/tex]
- Numerator: [tex]\(1 - (-279936) = 1 + 279936 = 279937\)[/tex]
- Substituting back:
[tex]\[
S_7 = -4 \times \frac{279937}{7}
\][/tex]
6. Final calculation:
- Compute [tex]\(\frac{279937}{7} = 39991\)[/tex], therefore,
[tex]\[
S_7 = -4 \times 39991 = -159964
\][/tex]
Thus, the sum of the geometric sequence for the first 7 terms is [tex]\(-159,964\)[/tex].
1. Identify the first term and common ratio:
- The first term ([tex]\(a_1\)[/tex]) of the sequence is [tex]\(-4\)[/tex].
- The common ratio ([tex]\(r\)[/tex]) can be found by dividing the second term by the first term. So,
[tex]\[
r = \frac{24}{-4} = -6
\][/tex]
2. Use the formula for the sum of a geometric sequence:
The sum of the first [tex]\(n\)[/tex] terms of a geometric sequence is given by the formula:
[tex]\[
S_n = a_1 \frac{1 - r^n}{1 - r}
\][/tex]
where:
- [tex]\(S_n\)[/tex] is the sum of the first [tex]\(n\)[/tex] terms,
- [tex]\(a_1\)[/tex] is the first term,
- [tex]\(r\)[/tex] is the common ratio,
- [tex]\(n\)[/tex] is the number of terms.
3. Substitute the known values into the formula:
- Here, [tex]\(a_1 = -4\)[/tex], [tex]\(r = -6\)[/tex], and [tex]\(n = 7\)[/tex].
Plug these values into the formula:
[tex]\[
S_7 = -4 \frac{1 - (-6)^7}{1 - (-6)}
\][/tex]
4. Calculate [tex]\( (-6)^7 \)[/tex]:
[tex]\[
(-6)^7 = -279936
\][/tex]
5. Calculate the sum:
- Denominator: [tex]\(1 - (-6) = 1 + 6 = 7\)[/tex]
- Numerator: [tex]\(1 - (-279936) = 1 + 279936 = 279937\)[/tex]
- Substituting back:
[tex]\[
S_7 = -4 \times \frac{279937}{7}
\][/tex]
6. Final calculation:
- Compute [tex]\(\frac{279937}{7} = 39991\)[/tex], therefore,
[tex]\[
S_7 = -4 \times 39991 = -159964
\][/tex]
Thus, the sum of the geometric sequence for the first 7 terms is [tex]\(-159,964\)[/tex].
