To determine whether the sequence [tex]\( \{f(n)\} \)[/tex] is an arithmetic sequence, we need to check if the differences between consecutive terms are constant. This constant difference, if it exists, is known as the common difference.
Given the sequence:
[tex]\[
\begin{array}{r|r|r|r|r}
n & 1 & 2 & 3 & 4 \\
\hline
f(n) & -4 & -8 & -12 & -16 \\
\end{array}
\][/tex]
Let's calculate the differences between consecutive terms:
1. Difference between [tex]\( f(2) \)[/tex] and [tex]\( f(1) \)[/tex]:
[tex]\[
-8 - (-4) = -8 + 4 = -4
\][/tex]
2. Difference between [tex]\( f(3) \)[/tex] and [tex]\( f(2) \)[/tex]:
[tex]\[
-12 - (-8) = -12 + 8 = -4
\][/tex]
3. Difference between [tex]\( f(4) \)[/tex] and [tex]\( f(3) \)[/tex]:
[tex]\[
-16 - (-12) = -16 + 12 = -4
\][/tex]
All the differences between consecutive terms are the same ([tex]\(-4\)[/tex]), hence the sequence is an arithmetic sequence.
The common difference is [tex]\(-4\)[/tex].
Thus, the answer is:
[tex]\[
\text{Yes}; -4
\][/tex]
