To solve this problem, let's break it down step by step:
1. Identify the Given Terms:
We are given:
- The 25th term of the arithmetic sequence ([tex]\(a_{25}\)[/tex]) is 0.
- The 60th term of the arithmetic sequence ([tex]\(a_{60}\)[/tex]) is -105.
2. Calculate the Common Difference:
The formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence is:
[tex]\[
a_n = a + (n - 1) \cdot d
\][/tex]
where:
- [tex]\(a\)[/tex] is the first term,
- [tex]\(d\)[/tex] is the common difference,
- [tex]\(n\)[/tex] is the position of the term in the sequence.
For the 25th term ([tex]\(a_{25}\)[/tex]):
[tex]\[
a_{25} = a + 24 \cdot d = 0
\][/tex]
For the 60th term ([tex]\(a_{60}\)[/tex]):
[tex]\[
a_{60} = a + 59 \cdot d = -105
\][/tex]
To find the common difference [tex]\(d\)[/tex], subtract the equation for the 25th term from the equation for the 60th term:
[tex]\[
(a + 59d) - (a + 24d) = -105 - 0
\][/tex]
Simplify this equation:
[tex]\[
35d = -105
\][/tex]
Solve for [tex]\(d\)[/tex]:
[tex]\[
d = \frac{-105}{35} = -3
\][/tex]
3. Calculate the First Term:
Now, use the value of [tex]\(d\)[/tex] to find the first term [tex]\(a\)[/tex]. Using the relation from the 25th term:
[tex]\[
0 = a + 24 \cdot (-3)
\][/tex]
Simplify and solve for [tex]\(a\)[/tex]:
[tex]\[
0 = a - 72 \implies a = 72
\][/tex]
4. Recursive Formula for the Sequence:
The recursive formula for the arithmetic sequence provides a way to find the [tex]\(n\)[/tex]th term based on the [tex]\((n-1)\)[/tex]th term. Since the common difference [tex]\(d\)[/tex] is -3, the recursive formula is:
[tex]\[
a(n) = a(n-1) + (-3)
\][/tex]
Or more simply:
[tex]\[
a(n) = a(n-1) - 3
\][/tex]
5. Formula for the [tex]\(n^{\text{th}}\)[/tex] Term:
Using the general formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence:
[tex]\[
a_n = a + (n - 1) \cdot d
\][/tex]
Substitute [tex]\(a = 72\)[/tex] and [tex]\(d = -3\)[/tex]:
[tex]\[
a_n = 72 + (n - 1) \cdot (-3)
\][/tex]
Simplify this formula:
[tex]\[
a_n = 72 - 3(n - 1) = 72 - 3n + 3 = 75 - 3n
\][/tex]
In summary:
- The first term [tex]\(a\)[/tex] is [tex]\(72\)[/tex].
- The common difference [tex]\(d\)[/tex] is [tex]\(-3\)[/tex].
- The recursive formula for the sequence is [tex]\(a(n) = a(n-1) - 3\)[/tex].
- The formula for the [tex]\(n\)[/tex]th term is [tex]\(a_n = 75 - 3n\)[/tex].
