# Can a arithmetic progression have a common difference of zero & a geometric progression have common ratio one?

How many three digit numbers have the property that their digits taken from left to right form an Arithmetic or Geometric Progression?

Eg. 123 is in form a AP when the digits are taken from left to right(1,2,3)

*should I count the numbers like 111,222.......999??

Because in my book they are not mentioned but

# Google says

"The common difference can be positive, negative or 'zero'. The English definition of the word 'progression' has nothing to do with the mathematical definition of arithmetic progression. Thus, a, a, a, a, a..... is a valid A.P. with c.d. 0.(It is also a geometric progression with common ratio=1."

• If you're asking (as per your title) whether you need to count "trivial" examples like $111, 222$ etc., I would say yes (although sometimes it's difficult to "mind-read" the examiner, so I might give both an answer without those and with those included for completeness and safety). If you're trying to get more detailed help in counting the possibilities, then you should clarify. Sep 16, 2015 at 3:16

## 1 Answer

The usual language convention in mathematics is to allow trivial cases (*) and to add adjectives such as "nontrivial", or in this case "nonconstant", when necessary to specifically exclude the trivia.

However, the words "progression of length $n$" nearly always means a nonconstant progression, and if that is not specified it is because the theorem is trivial, or trivially false, for constant progressions. Van der Waerden's theorem and its extensions are an example.

(*) if allowing the trivial cases does not require special clarifications in an annoyingly large proportion of theorems.