To find the [tex]\( n \)[/tex]th term of a geometric sequence, we 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]
where:
- [tex]\( a_1 \)[/tex] is the first term of the sequence,
- [tex]\( r \)[/tex] is the common ratio,
- [tex]\( n \)[/tex] is the term number.
Given values:
- The first term [tex]\( a_1 = 3 \)[/tex],
- The common ratio [tex]\( r = 2 \)[/tex].
By substituting these values into the general formula, we get:
[tex]\[ a_n = 3 \cdot 2^{n-1} \][/tex]
Let's verify if this formula matches one of the given options:
- [tex]\( a_n = 3 + 2(n-1) \)[/tex]
- [tex]\( a_n = 3(n-1) + 2 \)[/tex]
- [tex]\( a_n = 3^{n-1} \cdot 2 \)[/tex]
- [tex]\( a_n = 3 \cdot 2^{n-1} \)[/tex]
Among the provided options, the correct formula is:
[tex]\[ a_n = 3 \cdot 2^{n-1} \][/tex]
Thus, the formula that can be used to find the [tex]\( n \)[/tex]th term in the geometric sequence where [tex]\( a_1 = 3 \)[/tex] and [tex]\( r = 2 \)[/tex] is:
[tex]\[ a_n = 3 \cdot 2^{n-1} \][/tex]
