Which explains why the sequence 81, 3, 1/9, . . . is arithmetic or geometric?
This sequence is geometric because each term is obtained by multiplying the previous term by 1/3. To see this, note that 3/81 = 1/27 is the common ratio between the first two terms, and 1/9 รท (3/81) = 1/27 is the common ratio between the second and third terms. Therefore, this sequence is geometric.
To determine whether a sequence is arithmetic or geometric, we need to examine the relationship between consecutive terms in the sequence.
In an arithmetic sequence, each term is obtained by adding a fixed value (called the common difference) to the previous term. To check if the sequence 81, 3, 1/9, ... is arithmetic, we subtract consecutive terms to see if they are equal:
3 - 81 = -78
1/9 - 3 = -2 8/9
As you can see, the differences between consecutive terms are not the same. Since the differences are not constant, this sequence is not arithmetic.
In a geometric sequence, each term is obtained by multiplying the previous term by a fixed value (called the common ratio). To check if the sequence 81, 3, 1/9, ... is geometric, we divide consecutive terms to see if they are equal:
3 / 81 = 1/27
1/9 / 3 = 1/27
As you can see, the quotients between consecutive terms are the same (1/27). Since the quotients are constant, this sequence is geometric.
Therefore, the sequence 81, 3, 1/9, ... is geometric.
To determine whether the given sequence 81, 3, 1/9, . . . is arithmetic or geometric, we will analyze the differences or ratios between consecutive terms.
Arithmetic Sequence:
An arithmetic sequence is a sequence where the difference between consecutive terms is constant.
Let's check the differences between consecutive terms:
3 - 81 = -78
1/9 - 3 = -8/9
Since the differences are not constant, the given sequence is not an arithmetic sequence.
Geometric Sequence:
A geometric sequence is a sequence where the ratio between consecutive terms is constant.
Let's check the ratios between consecutive terms:
3 / 81 = 1/27
(1/9) / 3 = 1/27
Here, we see that the ratios are equal to 1/27, which is a constant.
Therefore, the given sequence 81, 3, 1/9, . . . is a geometric sequence.