# Is the same term geometric or arithmetic?

Is a series with the same number for every term a geometric or arithmetic series?

• Isn't it both? The common difference is $0$ and the common ratio is $1$!
– user441558
Jun 17 '17 at 3:29
• Or is it neither? Jun 17 '17 at 3:31
• Well, @ellieff1493, whether it's "both" or "neither" is really just going to depend on the precise definition of arithmetic or geometric progression that you use. Jun 17 '17 at 3:34
• @ellieff1493 Why is it neither? I know you can't employ the formula for the sum of terms of geometric series $S(n)=a(1-r^n)/(1-r)$ if $r=1$ but that does not mean a sequence like $1,1,1,1,...$ is not geometric!
– user441558
Jun 17 '17 at 3:36
• And an explanation I found is that arithmetic are linear but geometric aren't Jun 17 '17 at 3:36

• Except when the series is exactly $0$, the common ratio does not exist. Jul 16 '17 at 1:19
• @FrenzyLi Think of geometric series in terms of multiplication, instead of division. You'll see that $0,0,0,\dots$ is not very diferent than $2,2,2,\dots$. Jul 16 '17 at 15:19