Arithmetic progression with complex common difference?

Suppose we have the following sequence:

$$\{0,i,2i,3i,4i,5i\}$$

Can we call this sequence an arithmetic progression with first term $0$ and common difference of $i$ ?

Clarification: Here, $i$ is referring to the imaginary unit, i.e., $i=\sqrt{-1}$

In general, I want to know if the common difference of an AP can be any complex value and not just real value.

Thanks!

Answer

You can define an arithmetic progression in any monoid $(M,+)$. It is then defined by a starting element $a\in M$ and an increment $b\in M$ and the recursion

$$a_0 = a\\ a_{n+1} = a_n + b$$

There is no reason to restrict to reals $(\mathbb R,+)$ or complex numbers $(\mathbb C, +)$. For some results about arithmetic progressions, you might want $M$ to be an (abelian) group or even a field (both is true for the two settings mentioned here).

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For a complex finite arithmetic progression $\{z,z+w, \ldots, z+nw\}$ to have a real sum, you must actually force

$$\Im \sum_{k=0}^n (z+kw) = \Im \left((n+1)z + \frac{n(n+1)}2w\right) = (n+1)\Im z + \frac{n(n+1)}2 \Im w \stackrel!=0$$
In other words you can freely pick the real parts of $z$ and $w$, but the imaginary parts must be related by
$$\Im w = - \frac2n \Im z$$
for some $n\in\mathbb N$ wich will double as the number of terms minus one (since we sum from $k=0$ to $n$, wich has $n+1$ summands)