Essentially, there is no practical difference between the terms "arithmetic progression" and "arithmetic sequence." They are used interchangeably to describe the same mathematical concept.
While some sources may suggest subtle distinctions, these are rarely enforced in practice. The distinction often revolves around whether a progression must have a specific formula to calculate its nth term, whereas a sequence can be any ordered list of numbers following a logical rule. However, this is not a universally accepted difference.
Here's a breakdown of why they're generally considered the same:
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Definition: Both arithmetic sequences and arithmetic progressions refer to a sequence of numbers such that the difference between the consecutive terms is constant. This constant difference is called the "common difference."
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Formula: The nth term (an) of both an arithmetic sequence and an arithmetic progression can be calculated using the formula: an = a1 + (n - 1)d, where a1 is the first term, n is the term number, and d is the common difference.
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Usage: In most mathematical contexts, "arithmetic sequence" and "arithmetic progression" are used synonymously.
Example:
The set of numbers {2, 4, 6, 8, 10...} can be correctly described as both an arithmetic sequence and an arithmetic progression. The common difference is 2, and the nth term can be expressed as 2n.
In summary:
While a theoretical distinction could be made, in practice and in mathematics education, "arithmetic sequence" and "arithmetic progression" are treated as identical terms that describe an ordered list of numbers with a constant difference between successive terms. Therefore, you can use either term without changing the meaning.
