Certainly! Let's identify the third term ([tex]\(a_3\)[/tex]) of the given sequence: [tex]\(0.25, 0.5, 0.75, 1, 1.25, 1.5, \ldots\)[/tex].
1. Identify the sequence type: The given sequence [tex]\(0.25, 0.5, 0.75, 1, 1.25, 1.5, \ldots\)[/tex] is an arithmetic sequence because the difference between consecutive terms is constant.
2. Determine the common difference:
[tex]\[
\text{Common difference } (d) = 0.5 - 0.25 = 0.25
\][/tex]
3. Formulate the general term of the arithmetic sequence: The general term of an arithmetic sequence can be written as:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
where [tex]\(a_n\)[/tex] is the [tex]\(n\)[/tex]-th term, [tex]\(a_1\)[/tex] is the first term, and [tex]\(d\)[/tex] is the common difference.
4. Substitute the known values:
- First term ([tex]\(a_1\)[/tex]) = [tex]\(0.25\)[/tex]
- Common difference ([tex]\(d\)[/tex]) = [tex]\(0.25\)[/tex]
- We need the third term ([tex]\(a_3\)[/tex]), so [tex]\(n = 3\)[/tex]
5. Calculate the third term:
[tex]\[
a_3 = a_1 + (3 - 1) \cdot d
\][/tex]
Substituting the values:
[tex]\[
a_3 = 0.25 + (3 - 1) \cdot 0.25
\][/tex]
[tex]\[
a_3 = 0.25 + 2 \cdot 0.25
\][/tex]
[tex]\[
a_3 = 0.25 + 0.5
\][/tex]
[tex]\[
a_3 = 0.75
\][/tex]
Thus, the third term [tex]\(a_3\)[/tex] of the sequence is:
[tex]\[
\boxed{0.75}
\][/tex]
