To find the 25th term of the given sequence [tex]\( 2, 10, 18, 26, 34, \ldots \)[/tex], we first identify the pattern and structure of the sequence. This sequence is an arithmetic progression (AP), characterized by a constant difference between consecutive terms.
Key components of the sequence:
- First term ([tex]\(a\)[/tex]): The first term of the sequence is [tex]\( a = 2 \)[/tex].
- Common difference ([tex]\(d\)[/tex]): The difference between the first two terms is [tex]\( 10 - 2 = 8 \)[/tex].
In an arithmetic progression, the [tex]\(n\)[/tex]-th term ([tex]\(a_n\)[/tex]) can be given by the formula:
[tex]\[ a_n = a + (n-1) \cdot d \][/tex]
Here, we need to find the 25th term ([tex]\(a_{25}\)[/tex]). Substituting the known values into the formula:
1. [tex]\(a = 2\)[/tex]
2. [tex]\(d = 8\)[/tex]
3. [tex]\(n = 25\)[/tex]
Now, substitute these values into the formula:
[tex]\[ a_{25} = 2 + (25-1) \times 8 \][/tex]
First, calculate the term inside the parenthesis:
[tex]\[ 25 - 1 = 24 \][/tex]
Next, multiply this result by the common difference ([tex]\(d\)[/tex]):
[tex]\[ 24 \times 8 = 192 \][/tex]
Finally, add this result to the first term ([tex]\(a\)[/tex]):
[tex]\[ 2 + 192 = 194 \][/tex]
Therefore, the 25th term of the sequence is:
[tex]\[ \boxed{194} \][/tex]
