In an arithmetic sequence with a common difference of 0, all the terms are identical.
Here's a breakdown:
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Definition of Arithmetic Sequence: An arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is constant. This constant difference is called the common difference.
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Common Difference (d) = 0: When the common difference, denoted by d, is equal to 0, it means that we are adding 0 to each term to get the next term.
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Implication for Terms: If we start with a first term, a, and add 0 repeatedly, the sequence will be a, a, a, a,...
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Example: Let's say the first term a is 5 and the common difference d is 0. The arithmetic sequence would be: 5, 5, 5, 5, 5,...
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Formula: The general formula for the nth term of an arithmetic sequence is:
- an = a + (n-1)d
If d = 0, then:
- an = a + (n-1)(0)
- an = a
This confirms that every term (an) is equal to the first term (a).
In summary, an arithmetic sequence with a common difference of zero results in a sequence where every term is the same value as the first term. This can be thought of as a constant sequence.
