To find the 20th term of the arithmetic sequence [tex]\(100, 96, 92, 88, \ldots\)[/tex], we can follow these steps:
1. Identify the first term ([tex]\(a_1\)[/tex]): The first term of the sequence is [tex]\(100\)[/tex].
2. Determine the common difference ([tex]\(d\)[/tex]): The common difference can be found by subtracting any term from the preceding term. For example:
[tex]\[
d = 96 - 100 = -4
\][/tex]
3. Use the formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence: The formula to find the [tex]\(n\)[/tex]th term ([tex]\(a_n\)[/tex]) is:
[tex]\[
a_n = a_1 + (n - 1)d
\][/tex]
In this case, we want to find the 20th term ([tex]\(a_{20}\)[/tex]).
4. Substitute the known values into the formula:
[tex]\[
a_{20} = 100 + (20 - 1)(-4)
\][/tex]
5. Calculate the expression inside the parentheses:
[tex]\[
20 - 1 = 19
\][/tex]
6. Multiply the common difference by the result:
[tex]\[
19 \times (-4) = -76
\][/tex]
7. Add this product to the first term:
[tex]\[
a_{20} = 100 + (-76) = 100 - 76 = 24
\][/tex]
Therefore, the 20th term of the arithmetic sequence is [tex]\( \boxed{24} \)[/tex].
