To determine the [tex]\(13^\text{th}\)[/tex] term of the geometric sequence, we need to identify the pattern and use the formula for the [tex]\(n^\text{th}\)[/tex] term 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]) of the sequence is 1.
- The common ratio ([tex]\(r\)[/tex]) is the factor by which we multiply each term to get the next term. In this sequence:
- [tex]\(2 = 1 \times 2\)[/tex],
- [tex]\(4 = 2 \times 2\)[/tex],
- [tex]\(8 = 4 \times 2\)[/tex],
So, the common ratio [tex]\(r\)[/tex] is 2.
2. Form the general formula for the [tex]\(n^\text{th}\)[/tex] term:
The [tex]\(n^\text{th}\)[/tex] term ([tex]\(a_n\)[/tex]) of a geometric sequence is given by:
[tex]\[
a_n = a \cdot r^{n-1}
\][/tex]
3. Substitute the known values to find the [tex]\(13^\text{th}\)[/tex] term:
- Here, [tex]\(a = 1\)[/tex],
- [tex]\(r = 2\)[/tex],
- and we need to find [tex]\(a_{13}\)[/tex], where [tex]\(n = 13\)[/tex].
Substitute these values into the formula:
[tex]\[
a_{13} = 1 \cdot 2^{13-1}
\][/tex]
4. Simplify the exponent:
[tex]\[
a_{13} = 1 \cdot 2^{12}
\][/tex]
5. Calculate [tex]\(2^{12}\)[/tex]:
[tex]\[
2^{12} = 4096
\][/tex]
6. Find the [tex]\(13^\text{th}\)[/tex] term:
[tex]\[
a_{13} = 1 \cdot 4096 = 4096
\][/tex]
Thus, the [tex]\(13^\text{th}\)[/tex] term of the sequence [tex]\(1, 2, 4, 8, \ldots\)[/tex] is:
[tex]\[
\boxed{4096}
\][/tex]
