Answer :
Sure, let's find the 9th term of the given geometric sequence: [tex]\(9x^7, -18x^9, 36x^{11}, \ldots\)[/tex].
1. Identify the terms and ratio:
- The first term ([tex]\(a_1\)[/tex]) is [tex]\(9x^7\)[/tex].
- The second term ([tex]\(a_2\)[/tex]) is [tex]\(-18x^9\)[/tex].
2. Find the common ratio ([tex]\(r\)[/tex]):
- To find the common ratio, divide the second term by the first term:
[tex]\[
r = \frac{-18x^9}{9x^7} = -2x^2
\][/tex]
This means each term is obtained by multiplying the previous term by [tex]\(-2x^2\)[/tex].
3. Use the formula for the [tex]\(n\)[/tex]-th term of a geometric sequence:
- The formula for the [tex]\(n\)[/tex]-th term is given by:
[tex]\[
a_n = a_1 \cdot r^{n-1}
\][/tex]
4. Calculate the 9th term:
- Plug [tex]\(n = 9\)[/tex] into the formula to find [tex]\(a_9\)[/tex]:
[tex]\[
a_9 = 9x^7 \cdot \left(-2x^2\right)^{(9-1)}
\][/tex]
[tex]\[
a_9 = 9x^7 \cdot \left(-2x^2\right)^8
\][/tex]
5. Compute [tex]\(\left(-2x^2\right)^8\)[/tex]:
- Calculate [tex]\((-2)^8\)[/tex] and [tex]\((x^2)^8\)[/tex] separately:
[tex]\[
(-2)^8 = 256
\][/tex]
[tex]\[
(x^2)^8 = x^{16}
\][/tex]
6. Multiply these results by the first term:
- Substitute back into the formula:
[tex]\[
a_9 = 9x^7 \cdot 256x^{16}
\][/tex]
- Combining the coefficients and exponents:
[tex]\[
a_9 = 2304x^{23}
\][/tex]
So, the 9th term of the sequence is [tex]\(2304x^{23}\)[/tex].
1. Identify the terms and ratio:
- The first term ([tex]\(a_1\)[/tex]) is [tex]\(9x^7\)[/tex].
- The second term ([tex]\(a_2\)[/tex]) is [tex]\(-18x^9\)[/tex].
2. Find the common ratio ([tex]\(r\)[/tex]):
- To find the common ratio, divide the second term by the first term:
[tex]\[
r = \frac{-18x^9}{9x^7} = -2x^2
\][/tex]
This means each term is obtained by multiplying the previous term by [tex]\(-2x^2\)[/tex].
3. Use the formula for the [tex]\(n\)[/tex]-th term of a geometric sequence:
- The formula for the [tex]\(n\)[/tex]-th term is given by:
[tex]\[
a_n = a_1 \cdot r^{n-1}
\][/tex]
4. Calculate the 9th term:
- Plug [tex]\(n = 9\)[/tex] into the formula to find [tex]\(a_9\)[/tex]:
[tex]\[
a_9 = 9x^7 \cdot \left(-2x^2\right)^{(9-1)}
\][/tex]
[tex]\[
a_9 = 9x^7 \cdot \left(-2x^2\right)^8
\][/tex]
5. Compute [tex]\(\left(-2x^2\right)^8\)[/tex]:
- Calculate [tex]\((-2)^8\)[/tex] and [tex]\((x^2)^8\)[/tex] separately:
[tex]\[
(-2)^8 = 256
\][/tex]
[tex]\[
(x^2)^8 = x^{16}
\][/tex]
6. Multiply these results by the first term:
- Substitute back into the formula:
[tex]\[
a_9 = 9x^7 \cdot 256x^{16}
\][/tex]
- Combining the coefficients and exponents:
[tex]\[
a_9 = 2304x^{23}
\][/tex]
So, the 9th term of the sequence is [tex]\(2304x^{23}\)[/tex].
