Can an arithmetic progression have a constant difference of 0?

Question

Can an arithmetic progression have a constant difference of 0?
like 0,0,0,0,0,0
or 1,1,1,1,1

ok thanks, just want to triple confirm

Answer 1

yes it can
but it will be a trivial sequence

Answer 2

Absolutely; the only thing that matters in an arithmetic progression is that the difference of any two consecutive terms are equal.

For instance:
1, 2, 3, 4, 5

(5 - 4) = (4 - 3) = (3 - 2) = (2 - 1) = 1

0, 0, 0, 0, 0, ...
(0 - 0) = (0 - 0) = (0 - 0) = (0 - 0) = 0

One thing to note about sequences which repeat the same value over and over is that they qualify as both an arithmetic progression (d = 0) and a geometric progression (r = 1, for the sequence having non-zero values).

Answer 3

All the formulas for an arithmetic progression will work even if the difference is zero (What is the nth term of the progression? What is the sum of n terms of the progression? etc.)

Answer 4

Wouldn't call that progression in the English sense of the word, but you could call a point a degenerate circle with radius equal to zero. Sounds like the same sort of idea.

Answer 5

Yes - but these kinds of arithmetic progressions don't get invited to many parties, if you know what I mean.

Answer 6

Well, I'm not a mathematician, so my answer depends on the definition of "progression". To me, the word implies progress, or movement, so, just on that basis, of the examples you gave, progress does not appear to occur, except in the length of your "progression".

I'd vote "NO".

Answer 7

yes if your progression's logic is multiplying the first by 1.