Certainly! Let's determine the explicit formula for the given arithmetic sequence.
The sequence given is:
[tex]\[ 9, 1, -7, -15, \ldots \][/tex]
In an arithmetic sequence, the difference between successive terms is constant. This difference is called the common difference.
1. First, let's identify the common difference (\( d \)):
The difference between the second term (1) and the first term (9) is:
[tex]\[ 1 - 9 = -8 \][/tex]
So, the common difference \( d \) is:
[tex]\[ d = -8 \][/tex]
2. The first term of the sequence is denoted as \( a_1 \):
[tex]\[ a_1 = 9 \][/tex]
3. The explicit formula for the \( n \)-th term (\( a_n \)) of an arithmetic sequence can be derived using the following formula:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Where:
- \( a_1 \) is the first term of the sequence.
- \( d \) is the common difference.
- \( n \) is the position of the term in the sequence.
4. Substituting the values of \( a_1 \) and \( d \) into the formula:
[tex]\[ a_n = 9 + (n - 1) \cdot (-8) \][/tex]
Therefore, the explicit formula for the sequence is:
[tex]\[ a_n = 9 - 8(n - 1) \][/tex]
or, simplified:
[tex]\[ a_n = 9 - 8n + 8 \][/tex]
[tex]\[ a_n = 17 - 8n \][/tex]
So, the explicit formula for the sequence is:
[tex]\[ a_n = 17 - 8n \][/tex]
