To determine which term of the arithmetic series [tex]\(5, 9, 13, \ldots\)[/tex] is equal to 85, we need to use the formula for the [tex]\(n\)[/tex]-th term of an arithmetic series. The formula is given by:
[tex]\[ a_n = a + (n-1) \cdot d \][/tex]
Here:
- [tex]\(a\)[/tex] is the first term of the series.
- [tex]\(d\)[/tex] is the common difference between the terms.
- [tex]\(a_n\)[/tex] is the [tex]\(n\)[/tex]-th term that we need to find.
Given the series [tex]\(5, 9, 13, \ldots\)[/tex]:
- The first term [tex]\(a = 5\)[/tex].
- The common difference [tex]\(d = 9 - 5 = 4\)[/tex].
- We need to find the term number [tex]\(n\)[/tex] such that [tex]\(a_n = 85\)[/tex].
Substitute the known values into the formula:
[tex]\[ 85 = 5 + (n-1) \cdot 4 \][/tex]
Now, solve for [tex]\(n\)[/tex]:
1. Subtract 5 from both sides:
[tex]\[ 85 - 5 = (n-1) \cdot 4 \][/tex]
[tex]\[ 80 = (n-1) \cdot 4 \][/tex]
2. Divide both sides by 4:
[tex]\[ \frac{80}{4} = n-1 \][/tex]
[tex]\[ 20 = n-1 \][/tex]
3. Add 1 to both sides:
[tex]\[ 20 + 1 = n \][/tex]
[tex]\[ n = 21 \][/tex]
Therefore, the 21st term of the series is 85.
