Identify the type of sequence and the common difference or ratio: 5, 10, 20, 40,...
Understand the Problem
The question is asking to identify whether the given sequence is an arithmetic or geometric sequence and to determine the common difference (for arithmetic) or common ratio (for geometric) of the sequence 5, 10, 20, 40, ... In this case, the sequence appears to be geometric as each term is multiplied by 2 to get the next term.
Answer
The sequence is geometric with a common ratio of $r = 2$.
Answer for screen readers
The sequence is geometric with a common ratio of $r = 2$.
Steps to Solve
- Identify the type of sequence
To determine if the sequence is arithmetic or geometric, we must look at how the terms are generated. An arithmetic sequence has a common difference, while a geometric sequence has a common ratio.
- Calculate the ratio between consecutive terms
For the sequence 5, 10, 20, 40, we find the ratio by dividing each term by the previous term:
-
From 5 to 10: $$ \text{Ratio} = \frac{10}{5} = 2 $$
-
From 10 to 20: $$ \text{Ratio} = \frac{20}{10} = 2 $$
-
From 20 to 40: $$ \text{Ratio} = \frac{40}{20} = 2 $$
- Conclude based on the ratio
Since the ratio between all consecutive terms is the same (2), this confirms that the sequence is geometric.
- Identify the common ratio
The common ratio of the sequence is 2, as determined from the calculations above.
The sequence is geometric with a common ratio of $r = 2$.
More Information
In a geometric sequence, each term is derived by multiplying the previous term by a constant factor, known as the common ratio. The common ratio defines the relationship between consecutive terms in the sequence, which in this case is consistently 2.
Tips
A common mistake is to confuse the common difference with the common ratio. Remember, a sequence is arithmetic if it has a constant addition (common difference) between terms and geometric if it has a constant multiplication (common ratio).
AI-generated content may contain errors. Please verify critical information
