Identify when a sequence is arithmetic.
Understand the Problem
The question is asking to identify the conditions under which a sequence is classified as arithmetic, specifically looking for the definition involving either the difference or ratio of terms in the sequence.
Answer
The sequence is arithmetic when the difference between any two consecutive terms remains constant.
Answer for screen readers
The sequence is arithmetic when the difference between any two consecutive terms remains constant.
Steps to Solve
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Understanding Arithmetic Sequences
An arithmetic sequence is defined as a sequence of numbers in which the difference between any two consecutive terms is constant. This means that if you take any two consecutive terms, subtract the first from the second, the result will always be the same. -
Identifying the Correct Option
From the options given, we need to identify which one correctly describes the property of an arithmetic sequence. -
Analyzing the Options
- The first option states that the difference is changing, which contradicts the definition of an arithmetic sequence.
- The second option refers to the ratio changing, which is unrelated to arithmetic sequences (it's more related to geometric sequences).
- The third option again talks about the ratio remaining constant, which is not applicable to arithmetic sequences.
- The fourth option describes the difference between terms remaining constant, which aligns perfectly with the definition of an arithmetic sequence.
The sequence is arithmetic when the difference between any two consecutive terms remains constant.
More Information
An arithmetic sequence can be expressed in the form $a_n = a_1 + (n - 1)d$, where $d$ is the common difference between the terms. The sequence 2, 4, 6, 8 is an example, where the common difference $d = 2$.
Tips
- Confusing arithmetic sequences with geometric sequences, which involve a constant ratio between terms rather than a constant difference.
- Misinterpreting the definition of constancy (e.g., thinking that changing differences can still yield an arithmetic sequence).
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