Determine whether the sequence is arithmetic, geometric, both, or neither.
16, 8, 4, 2, . . .
A. arithmetic
B. geometric
C. neither
D. both
B. geometric
To determine whether the given sequence is arithmetic, geometric, both, or neither, we need to examine the differences between consecutive terms.
In this sequence, if we subtract each term from its subsequent term, we get the following values:
8 - 16 = -8
4 - 8 = -4
2 - 4 = -2
The differences between consecutive terms are not constant, so the sequence is not arithmetic.
Now, let's examine the ratios between consecutive terms:
8 ÷ 16 = 0.5
4 ÷ 8 = 0.5
2 ÷ 4 = 0.5
The ratios between consecutive terms are constant (0.5), so the sequence is geometric.
Therefore, the correct answer is:
B. geometric
To determine whether a sequence is arithmetic, geometric, both, or neither, we need to understand the properties of each type of sequence.
1. Arithmetic sequence: In an arithmetic sequence, each term can be obtained by adding a common difference to the previous term. For example, 2, 4, 6, 8 is an arithmetic sequence with a common difference of 2.
2. Geometric sequence: In a geometric sequence, each term can be obtained by multiplying the previous term by a common ratio. For example, 2, 4, 8, 16 is a geometric sequence with a common ratio of 2.
Now let's analyze the given sequence: 16, 8, 4, 2, ...
To determine whether it is arithmetic or geometric, we can calculate the common difference (d) for arithmetic sequences and the common ratio (r) for geometric sequences.
For an arithmetic sequence, we subtract consecutive terms to find the common difference:
8 - 16 = -8
4 - 8 = -4
2 - 4 = -2
As we can see, the differences are not the same, indicating that this is not an arithmetic sequence.
For a geometric sequence, we divide consecutive terms to find the common ratio:
8 / 16 = 0.5
4 / 8 = 0.5
2 / 4 = 0.5
As we can see, the ratios are all the same (0.5), indicating that this is a geometric sequence.
Therefore, the given sequence 16, 8, 4, 2, ... is a geometric sequence.
The answer is B. geometric.