Let's find the 58th term of the arithmetic sequence [tex]\(1, 3, 5, \ldots\)[/tex].
First, let's identify the relevant characteristics of the arithmetic sequence:
- The first term ([tex]\(a_1\)[/tex]) is [tex]\(1\)[/tex].
- The common difference ([tex]\(d\)[/tex]) is [tex]\(3 - 1 = 2\)[/tex].
The formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
We are asked to find the 58th term, so we will substitute [tex]\(n = 58\)[/tex], [tex]\(a_1 = 1\)[/tex], and [tex]\(d = 2\)[/tex] into the formula.
Substitute the values into the formula:
[tex]\[
a_{58} = 1 + (58-1) \cdot 2
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
a_{58} = 1 + 57 \cdot 2
\][/tex]
Multiply:
[tex]\[
a_{58} = 1 + 114
\][/tex]
Add the terms:
[tex]\[
a_{58} = 115
\][/tex]
Therefore, the 58th term of the arithmetic sequence [tex]\(1, 3, 5, \ldots\)[/tex] is [tex]\(115\)[/tex].
