Sure, let's find an explicit formula for the arithmetic sequence 81, 54, 27, 0, ...
1. Identify the First Term ([tex]\(a_1\)[/tex]):
- The first term of the sequence is given as 81.
[tex]\[
a_1 = 81
\][/tex]
2. Determine the Common Difference ([tex]\(d\)[/tex]):
- To find the common difference, subtract the first term from the second term:
[tex]\[
d = 54 - 81 = -27
\][/tex]
3. Write the Formula for the [tex]\(n\)[/tex]-th Term:
- The general formula for the [tex]\(n\)[/tex]-th term of an arithmetic sequence is given by:
[tex]\[
a(n) = a_1 + (n - 1) \cdot d
\][/tex]
4. Substitute the Known Values:
- Substitute [tex]\(a_1 = 81\)[/tex] and [tex]\(d = -27\)[/tex] into the formula:
[tex]\[
a(n) = 81 + (n - 1) \cdot (-27)
\][/tex]
5. Simplify the Expression:
- Distribute the [tex]\(-27\)[/tex] in the formula:
[tex]\[
a(n) = 81 - 27(n - 1)
\][/tex]
- Distribute and simplify further:
[tex]\[
a(n) = 81 - 27n + 27
\][/tex]
[tex]\[
a(n) = 108 - 27n
\][/tex]
So, the explicit formula for the given arithmetic sequence is:
[tex]\[
a(n) = 108 - 27n
\][/tex]
This formula allows you to find the [tex]\(n\)[/tex]-th term of the sequence by substituting [tex]\(n\)[/tex] into the formula.
