To determine the correct formula to find the [tex]$n$[/tex]th term in a geometric sequence where the first term [tex]\( a_1 = 3 \)[/tex] and the common ratio [tex]\( r = 2 \)[/tex], we need to use the general formula for the [tex]$n$[/tex]th term of a geometric sequence:
[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]
Here's the step-by-step process to confirm which given option matches this formula:
1. Identify the first term and common ratio:
- First term ([tex]\(a_1\)[/tex]): 3
- Common ratio ([tex]\(r\)[/tex]): 2
2. Substitute these values into the general formula:
- General formula for the [tex]$n$[/tex]th term is [tex]\( a_n = a_1 \cdot r^{n-1} \)[/tex].
- Substitute [tex]\( a_1 = 3 \)[/tex] and [tex]\( r = 2 \)[/tex]:
[tex]\[
a_n = 3 \cdot 2^{n-1}
\][/tex]
3. Compare this with the given options:
- [tex]\( a_n = 3 + 2(n-1) \)[/tex] (Option 1): This is an arithmetic sequence formula, not geometric.
- [tex]\( a_n = 3(n-1) + 2 \)[/tex] (Option 2): This is also not correct for a geometric sequence.
- [tex]\( a_n = 3^{n-1} \cdot 2 \)[/tex] (Option 3): This formula incorrectly places the base for exponentiation.
- [tex]\( a_n = 3 \cdot 2^{n-1} \)[/tex] (Option 4): This matches our derived formula exactly.
Thus, the correct formula to find the [tex]$n$[/tex]th term in this geometric sequence is:
[tex]\[
a_n = 3 \cdot 2^{n-1}
\][/tex]
So, the correct option is:
[tex]\[
4
\][/tex]
