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!
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).
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)