To find the specified [tex]\(n\)[/tex]th term of a geometric sequence, we can use the following information given in the problem:
- The 4th term ([tex]\(a_4\)[/tex]) is 10.
- The common ratio ([tex]\(r\)[/tex]) is 2.
- We need to find the 9th term ([tex]\(a_9\)[/tex]).
The general formula to find the [tex]\(n\)[/tex]th term of a geometric sequence is:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
where [tex]\(a_1\)[/tex] is the first term of the sequence, [tex]\(r\)[/tex] is the common ratio, and [tex]\(n\)[/tex] is the term number.
Given:
[tex]\[
a_4 = 10 \quad \text{and} \quad r = 2
\][/tex]
First, we need to find the first term of the sequence [tex]\(a_1\)[/tex]. Using the formula for the 4th term:
[tex]\[
a_4 = a_1 \cdot r^{(4-1)}
\][/tex]
Plugging in the known values:
[tex]\[
10 = a_1 \cdot 2^3
\][/tex]
Solving for [tex]\(a_1\)[/tex]:
[tex]\[
10 = a_1 \cdot 8 \implies a_1 = \frac{10}{8} = 1.25
\][/tex]
Now that we have the first term [tex]\(a_1 = 1.25\)[/tex], we can find the 9th term ([tex]\(a_9\)[/tex]) using the formula for the [tex]\(n\)[/tex]th term of a geometric sequence:
[tex]\[
a_9 = a_1 \cdot r^{(9-1)}
\][/tex]
Plugging in the values:
[tex]\[
a_9 = 1.25 \cdot 2^8
\][/tex]
Calculating the power of 2:
[tex]\[
2^8 = 256
\][/tex]
Thus:
[tex]\[
a_9 = 1.25 \cdot 256 = 320
\][/tex]
Therefore, the 9th term ([tex]\(a_9\)[/tex]) of the geometric sequence is [tex]\(\boxed{320}\)[/tex].
