To find the 20th term ([tex]\(a_{20}\)[/tex]) of the given arithmetic sequence:
1. Identify the first term ([tex]\(a_1\)[/tex]) and the common difference ([tex]\(d\)[/tex]) in the sequence.
From the sequence [tex]\(-7, -4, -1, 2, 5, \ldots\)[/tex]:
- The first term [tex]\(a_1 = -7\)[/tex].
- The common difference [tex]\(d\)[/tex] can be found by subtracting the first term from the second term:
[tex]\[
d = -4 - (-7) = 3
\][/tex]
2. Use the formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
Here, we are looking for the 20th term ([tex]\(a_{20}\)[/tex]). Substitute [tex]\(a_1 = -7\)[/tex], [tex]\(d = 3\)[/tex], and [tex]\(n = 20\)[/tex] into the formula:
[tex]\[
a_{20} = -7 + (20 - 1) \cdot 3
\][/tex]
3. Simplify the expression inside the parentheses and perform the multiplication:
[tex]\[
a_{20} = -7 + 19 \cdot 3
\][/tex]
[tex]\[
19 \cdot 3 = 57
\][/tex]
4. Finally, add the result to the first term:
[tex]\[
a_{20} = -7 + 57
\][/tex]
[tex]\[
a_{20} = 50
\][/tex]
Therefore, the 20th term of the sequence [tex]\(a_{20}\)[/tex] is [tex]\(50\)[/tex].
