Let's analyze the given sequence step by step to determine which statement is correct:
The given sequence is:
[tex]\[ 5, 2, -3, -10, -19 \][/tex]
To determine whether the sequence is arithmetic, we need to calculate the differences between each pair of consecutive terms. An arithmetic sequence has a constant common difference between each consecutive term.
1. Calculate the difference between the first and second terms:
[tex]\[ 2 - 5 = -3 \][/tex]
So, the difference between the first and second terms is [tex]\(-3\)[/tex].
2. Calculate the difference between the second and third terms:
[tex]\[ -3 - 2 = -5 \][/tex]
So, the difference between the second and third terms is [tex]\(-5\)[/tex].
3. Calculate the difference between the third and fourth terms:
[tex]\[ -10 - (-3) = -10 + 3 = -7 \][/tex]
So, the difference between the third and fourth terms is [tex]\(-7\)[/tex].
4. Calculate the difference between the fourth and fifth terms:
[tex]\[ -19 - (-10) = -19 + 10 = -9 \][/tex]
So, the difference between the fourth and fifth terms is [tex]\(-9\)[/tex].
The differences between the consecutive terms are:
[tex]\[ -3, -5, -7, -9 \][/tex]
Since the differences are not consistent, there is no common difference. Therefore, the sequence is not arithmetic.
So, the correct statement is:
"The sequence is not arithmetic because there is no common difference. The differences are [tex]\(-3, -5, -7\)[/tex] and [tex]\(-9\)[/tex]."
