To find the 25th term, [tex]\( a_{25} \)[/tex], of the arithmetic sequence given by [tex]\( 12, 20, 28, \ldots \)[/tex], we can follow these steps:
1. Identify the first term ([tex]\( a_1 \)[/tex]) and the common difference ([tex]\( d \)[/tex]):
- The first term, [tex]\( a_1 \)[/tex], is 12.
- The common difference, [tex]\( d \)[/tex], is the difference between consecutive terms.
[tex]\[
d = 20 - 12 = 8
\][/tex]
2. Recall the formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence:
[tex]\[
a_n = a_1 + (n - 1)d
\][/tex]
Here, [tex]\( a_n \)[/tex] is the [tex]\( n \)[/tex]-th term, [tex]\( a_1 \)[/tex] is the first term, [tex]\( d \)[/tex] is the common difference, and [tex]\( n \)[/tex] is the term number.
3. Substitute the known values into the formula to find the 25th term:
[tex]\[
a_{25} = 12 + (25 - 1) \cdot 8
\][/tex]
[tex]\[
a_{25} = 12 + 24 \cdot 8
\][/tex]
4. Perform the arithmetic operations step-by-step:
[tex]\[
24 \cdot 8 = 192
\][/tex]
[tex]\[
a_{25} = 12 + 192 = 204
\][/tex]
Therefore, the 25th term of the sequence is [tex]\( \boxed{204} \)[/tex].
