The question, as written, presents an issue because the given sequence 2, 4, 8, 16, 32, 64, 2, 662 is not a geometric sequence with a single common ratio throughout. A geometric sequence has a constant ratio between each term. The terms 2, 4, 8, 16, 32, and 64 form a geometric sequence, but the terms '2' and '662' do not fit this pattern. To accurately answer the question, we need to clarify what segment of the sequence the question is referring to. We can rephrase the question as follows:
Revised Question: What is the common ratio between the successive terms in the geometric sequence contained within the sequence 2, 4, 8, 16, 32, 64, 2, 662?
Given the revised question, we are now looking for the common ratio within the initial geometric part of the sequence: 2, 4, 8, 16, 32, 64.
The common ratio between successive terms in this initial geometric part of the sequence is 2.
Understanding Common Ratio
A common ratio is the factor by which each term in a geometric sequence is multiplied to get the next term.
Calculating the Common Ratio
We can determine this by dividing any term by its preceding term. Let's test this within the initial geometric sequence segment:
- 4 / 2 = 2
- 8 / 4 = 2
- 16 / 8 = 2
- 32 / 16 = 2
- 64 / 32 = 2
The consistent result of '2' confirms that the common ratio for the first portion of the sequence is 2.
Reference Information
As shown in the reference material provided, the sequence 2, -4, 8, -16, 32, -64, … has a common ratio of -2. This is because each term is multiplied by -2 to get the next term.
| Sequence | Common Ratio |
|---|---|
| 2, 4, 8, 16, 32, 64, ... | 2 |
| 2, -4, 8, -16, 32, -64, ... | -2 |
In summary
The sequence 2, 4, 8, 16, 32, 64 has a common ratio of 2. This is derived from the first six terms of the given sequence. The last two terms 2 and 662, do not conform to this pattern, therefore the whole sequence 2, 4, 8, 16, 32, 64, 2, 662 does not have a single common ratio.
