Let's analyze the given arithmetic sequence: [tex]\(-9, -2, 5, 12, \ldots\)[/tex].
First, we need to determine [tex]\(f(1)\)[/tex], which is the first term of the sequence. From the given sequence, the first term is [tex]\(-9\)[/tex].
[tex]\[ f(1) = -9 \][/tex]
Next, we need to find the common difference of the arithmetic sequence. The common difference ([tex]\(d\)[/tex]) in an arithmetic sequence is the difference between any two consecutive terms. Let's compute it using the first two terms:
Given terms:
[tex]\[ a_1 = -9 \][/tex]
[tex]\[ a_2 = -2 \][/tex]
The common difference [tex]\(d\)[/tex] is:
[tex]\[ d = a_2 - a_1 = -2 - (-9) \][/tex]
[tex]\[ d = -2 + 9 \][/tex]
[tex]\[ d = 7 \][/tex]
So, the values we have found are:
[tex]\[ f(1) = -9 \][/tex]
[tex]\[ \text{Common difference} = 7 \][/tex]
Therefore, the correct answer is:
A. [tex]\( f(1) = -9 \)[/tex]
Common difference : 7
