Answer :
To find the 12th term of the given geometric sequence [tex]\(-x^4, 3x^7, -9x^{10}, \ldots\)[/tex], we will follow these steps:
1. Identify the first term ([tex]\(a\)[/tex]) and the common ratio ([tex]\(r\)[/tex]):
- The first term, [tex]\(a\)[/tex], is [tex]\(-x^4\)[/tex].
2. Find the common ratio ([tex]\(r\)[/tex]):
- The second term is [tex]\(3x^7\)[/tex].
The common ratio, [tex]\(r\)[/tex], can be found by dividing the second term by the first term:
[tex]\[
r = \frac{3x^7}{-x^4} = -3x^{7-4} = -3x^3
\][/tex]
3. Use the formula for the [tex]\(n\)[/tex]-th term of a geometric sequence:
The formula to find the [tex]\(n\)[/tex]-th term of a geometric sequence is:
[tex]\[
a_n = a \cdot r^{(n-1)}
\][/tex]
Substituting the values for the 12th term ([tex]\(n = 12\)[/tex]):
[tex]\[
a_{12} = -x^4 \cdot (-3x^3)^{11}
\][/tex]
4. Simplify the expression:
First, calculate [tex]\((-3x^3)^{11}\)[/tex]:
[tex]\[
(-3x^3)^{11} = (-3)^{11} \cdot (x^3)^{11} = -177147 \cdot x^{33}
\][/tex]
Then, multiply by the first term:
[tex]\[
a_{12} = -x^4 \cdot (-177147 \cdot x^{33}) = 177147 \cdot x^{4+33} = 177147x^{37}
\][/tex]
5. Conclusion:
Therefore, the 12th term of the sequence is [tex]\(177147x^{37}\)[/tex].
1. Identify the first term ([tex]\(a\)[/tex]) and the common ratio ([tex]\(r\)[/tex]):
- The first term, [tex]\(a\)[/tex], is [tex]\(-x^4\)[/tex].
2. Find the common ratio ([tex]\(r\)[/tex]):
- The second term is [tex]\(3x^7\)[/tex].
The common ratio, [tex]\(r\)[/tex], can be found by dividing the second term by the first term:
[tex]\[
r = \frac{3x^7}{-x^4} = -3x^{7-4} = -3x^3
\][/tex]
3. Use the formula for the [tex]\(n\)[/tex]-th term of a geometric sequence:
The formula to find the [tex]\(n\)[/tex]-th term of a geometric sequence is:
[tex]\[
a_n = a \cdot r^{(n-1)}
\][/tex]
Substituting the values for the 12th term ([tex]\(n = 12\)[/tex]):
[tex]\[
a_{12} = -x^4 \cdot (-3x^3)^{11}
\][/tex]
4. Simplify the expression:
First, calculate [tex]\((-3x^3)^{11}\)[/tex]:
[tex]\[
(-3x^3)^{11} = (-3)^{11} \cdot (x^3)^{11} = -177147 \cdot x^{33}
\][/tex]
Then, multiply by the first term:
[tex]\[
a_{12} = -x^4 \cdot (-177147 \cdot x^{33}) = 177147 \cdot x^{4+33} = 177147x^{37}
\][/tex]
5. Conclusion:
Therefore, the 12th term of the sequence is [tex]\(177147x^{37}\)[/tex].
