To find the [tex]\( 12^{\text{th}} \)[/tex] term of the given geometric sequence:
[tex]\[ 3, 12, 48, 192, \ldots \][/tex]
we need to follow these steps:
1. Identify the first term [tex]\( a \)[/tex] and the common ratio [tex]\( r \)[/tex]:
- The first term [tex]\( a \)[/tex] is the first number in the sequence, which is 3.
- The common ratio [tex]\( r \)[/tex] is the factor by which each term is multiplied to get the next term. We can find [tex]\( r \)[/tex] by dividing the second term by the first term:
[tex]\[
r = \frac{12}{3} = 4
\][/tex]
2. Use the formula for the [tex]\( n \)[/tex]-th term of a geometric sequence:
The formula to find the [tex]\( n \)[/tex]-th term [tex]\( a_n \)[/tex] is given by:
[tex]\[
a_n = a \cdot r^{n-1}
\][/tex]
Where:
- [tex]\( a \)[/tex] is the first term.
- [tex]\( r \)[/tex] is the common ratio.
- [tex]\( n \)[/tex] is the term number we want to find.
3. Plug in the values for [tex]\( a \)[/tex], [tex]\( r \)[/tex], and [tex]\( n \)[/tex]:
We want to find the [tex]\( 12^{\text{th}} \)[/tex] term, so [tex]\( n = 12 \)[/tex]:
[tex]\[
a_{12} = 3 \cdot 4^{12-1}
\][/tex]
4. Calculate [tex]\( 4^{11} \)[/tex]:
[tex]\[
4^{11} = 4194304
\][/tex]
5. Multiply the first term by [tex]\( 4^{11} \)[/tex]:
[tex]\[
a_{12} = 3 \cdot 4194304 = 12582912
\][/tex]
Therefore, the [tex]\( 12^{\text{th}} \)[/tex] term of the geometric sequence is [tex]\( 12,582,912 \)[/tex].
