Certainly! Let's analyze the given sequence:
[tex]\[ 29, 25, 21, 17, 13 \][/tex]
First, we'll determine if there is a common difference between the terms, which would indicate that the sequence is arithmetic. We do this by subtracting each term from the one before it:
[tex]\[ 25 - 29 = -4 \][/tex]
[tex]\[ 21 - 25 = -4 \][/tex]
[tex]\[ 17 - 21 = -4 \][/tex]
[tex]\[ 13 - 17 = -4 \][/tex]
We can see that the difference between consecutive terms is [tex]\(-4\)[/tex]. Since the difference is constant, the sequence is indeed arithmetic, and the common difference ([tex]\(d\)[/tex]) is [tex]\(-4\)[/tex].
Next, let's use the general formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence:
[tex]\[ a_n = a_1 + (n-1) \cdot d \][/tex]
Where:
- [tex]\(a_n\)[/tex] is the [tex]\(n\)[/tex]th term,
- [tex]\(a_1\)[/tex] is the first term,
- [tex]\(d\)[/tex] is the common difference,
- [tex]\(n\)[/tex] is the term number.
In our sequence:
- The first term [tex]\(a_1\)[/tex] is [tex]\(29\)[/tex],
- The common difference [tex]\(d\)[/tex] is [tex]\(-4\)[/tex].
Now, let's substitute these values into the formula:
[tex]\[ a_n = 29 + (n-1) \cdot (-4) \][/tex]
Simplify the expression:
[tex]\[ a_n = 29 - 4(n-1) \][/tex]
[tex]\[ a_n = 29 - 4n + 4 \][/tex]
[tex]\[ a_n = 33 - 4n \][/tex]
Thus, the [tex]\(n\)[/tex]th term of the sequence can be expressed as:
[tex]\[ a_n = 33 - 4n \][/tex]
This is the expression for the [tex]\(n\)[/tex]th term of the given sequence.
