To find the 20th term of the given arithmetic sequence [tex]\(17, 13, 9, \ldots\)[/tex], we can use the formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence, which is:
[tex]\[ a_n = a_1 + (n-1)d \][/tex]
where:
- [tex]\(a_n\)[/tex] is the [tex]\(n\)[/tex]-th term we want to find,
- [tex]\(a_1\)[/tex] is the first term of the sequence,
- [tex]\(d\)[/tex] is the common difference between the terms,
- [tex]\(n\)[/tex] is the term number.
Let's break down the steps:
1. Identify the first term ([tex]\(a_1\)[/tex]):
The first term of the sequence is given as [tex]\(a_1 = 17\)[/tex].
2. Find the common difference ([tex]\(d\)[/tex]):
The common difference is the difference between any two consecutive terms of the sequence.
[tex]\[ d = a_2 - a_1 = 13 - 17 = -4 \][/tex]
3. Determine the term number ([tex]\(n\)[/tex]):
We are asked to find the 20th term, so [tex]\(n = 20\)[/tex].
4. Substitute the values into the formula:
[tex]\[
a_{20} = 17 + (20-1)(-4)
\][/tex]
Simplify the expression inside the parentheses first:
[tex]\[
a_{20} = 17 + 19(-4)
\][/tex]
5. Perform the multiplication:
[tex]\[
a_{20} = 17 + 19 \times (-4)
= 17 - 76
\][/tex]
6. Calculate the final result:
[tex]\[
a_{20} = 17 - 76 = -59
\][/tex]
Therefore, the 20th term of the arithmetic sequence [tex]\(17, 13, 9, \ldots\)[/tex] is [tex]\(-59\)[/tex].
