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I'm pretty sure I've heard this many times from people that I require at least $3$ terms to form an AP.

But now, after the reading the definition of AP,

An arithmetic progression(AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.

I'm having doubts here, if I'm following this definition correctly, then the difference between any two consecutive terms is constant in an AP.

Can I say, that even $2$ terms can form an AP?

Example : I could say $2,9$ are in AP or $6,15$ are in AP?

Thoughts?

William
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  • Two numbers already form a (trivial) arithmetic progression. The definition is that the differences are the same, and we only have one such difference. But this case is so boring that it makes sense to rule such cases out. – Peter Jun 18 '18 at 15:10
  • AP is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. – LAAE Jun 25 '18 at 22:01

2 Answers2

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According to wikipedia:

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with common difference of 2.

So an argument could be made that, technically, a null sequence (no numbers at all), as well as a single number or two numbers, would be a arithmetic progression, but they would be trivial examples that don't exhibit what people typically think of as an "arithmetic progression". There are probably cases where a definition is used that excludes these types, but this would not be a consistent situation.

Acccumulation
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Any two numbers are technically in an trivial AP because there is only one difference of numbers to consider. However, when there are three or more numbers, there are two or more differences that need to be held constant. This means that some actual pattern is being upheld.

bkarthik
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