Which explains why the sequence 64, 4, 1/4,.. is arithmetic or geometric
The sequence 64, 4, 1/4,.. is geometric because each term is obtained by dividing the previous term by 16, which is a constant ratio. Specifically, to get from 64 to 4, you divide by 16. To get from 4 to 1/4, you again divide by 16. Therefore, the common ratio between any two consecutive terms is always 1/16.
To determine if a sequence is arithmetic or geometric, we need to identify the relationship between its terms.
In an arithmetic sequence, each term is obtained by adding a constant difference to the previous term. So, if the sequence is arithmetic, there should be a common difference between consecutive terms.
In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio. If the sequence is geometric, there should be a common ratio between consecutive terms.
Now, let's analyze the given sequence: 64, 4, 1/4, ...
To check if it is an arithmetic sequence, we need to find the differences between consecutive terms:
4 - 64 = -60
1/4 - 4 = -15.75
As we can see, the differences are not constant. The first difference is -60, and the second difference is -15.75. Therefore, the sequence is not arithmetic.
To check if it is a geometric sequence, we need to find the ratios between consecutive terms:
4/64 = 1/16
(1/4)/(4) = 1/16
The ratios between consecutive terms are constant, and they are equal to 1/16. Hence, the sequence is geometric.
In conclusion, the sequence 64, 4, 1/4,.. is a geometric sequence because the ratios between consecutive terms are constant.
To determine whether the sequence 64, 4, 1/4,.. is arithmetic or geometric, we need to check if there is a common difference or a common ratio between the terms.
1. Arithmetic Sequence:
An arithmetic sequence is a sequence in which the difference between two consecutive terms is constant.
Let's check the difference between the terms:
4 - 64 = -60
1/4 - 4 = -15.75
Since the differences between the consecutive terms (-60 and -15.75) are not constant, the sequence is not arithmetic.
2. Geometric Sequence:
A geometric sequence is a sequence in which each term is obtained by multiplying the preceding term by a constant factor.
Let's check the ratio between the terms:
4 ÷ 64 = 0.0625
1/4 ÷ 4 = 0.0625
The ratio between the terms (0.0625) remains constant. Therefore, the sequence is geometric.
In conclusion, the sequence 64, 4, 1/4,.. is a geometric sequence.