To determine which of the sequences is not arithmetic, we must first understand what an arithmetic sequence is. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is called the common difference.
Let's analyze each sequence:
1. [tex]\( 15, 9, 3, -3, -9 \)[/tex]
To check for a common difference, we subtract each term from the following term:
[tex]\( 9 - 15 = -6 \)[/tex]
[tex]\( 3 - 9 = -6 \)[/tex]
[tex]\( -3 - 3 = -6 \)[/tex]
[tex]\( -9 - (-3) = -6 \)[/tex]
The common difference here is [tex]\(-6\)[/tex], so the sequence is arithmetic.
2. [tex]\( 2, 4, 8, 16, 32 \)[/tex]
For this sequence, we check again:
[tex]\( 4 - 2 = 2 \)[/tex]
[tex]\( 8 - 4 = 4 \)[/tex]
[tex]\( 16 - 8 = 8 \)[/tex]
[tex]\( 32 - 16 = 16 \)[/tex]
The differences here are [tex]\(2\)[/tex], [tex]\(4\)[/tex], [tex]\(8\)[/tex], and [tex]\(16\)[/tex], which are not constant. The sequence is doubling each time, which indicates that it is geometric, not arithmetic.
3. [tex]\( 4, 7, 10, 13, 16 \)[/tex]
We'll examine the differences between terms:
[tex]\( 7 - 4 = 3 \)[/tex]
[tex]\( 10 - 7 = 3 \)[/tex]
[tex]\( 13 - 10 = 3 \)[/tex]
[tex]\( 16 - 13 = 3 \)[/tex]
The common difference is [tex]\(3\)[/tex]. Since it is constant, the sequence is arithmetic.
4. [tex]\( 1, 2, 3, 4, 5 \)[/tex]
We calculate the differences:
[tex]\( 2 - 1 = 1 \)[/tex]
[tex]\( 3 - 2 = 1 \)[/tex]
[tex]\( 4 - 3 = 1 \)[/tex]
[tex]\( 5 - 4 = 1 \)[/tex]
The common difference is [tex]\(1\)[/tex]. Thus, the sequence is also arithmetic.
Conclusion:
After reviewing each sequence, we find that the second sequence, [tex]\(2, 4, 8, 16, 32\)[/tex], is not arithmetic because it does not have a constant difference between its terms. It is a geometric sequence where each term is multiplied by a common ratio to get the next term. In this case, the common ratio is [tex]\(2\)[/tex]. Thus, the sequence [tex]\(2, 4, 8, 16, 32\)[/tex] is the one that is NOT an arithmetic sequence.
