Arithmetic Progressions in Complex Variables From Stein and Shakarchi's Complex Analysis book, Chapter 1 Exercise 22 asks the following:

Let $\Bbb N=\{1,2,\ldots\}$ denote the set of positive integers. A subset $S\subseteq \Bbb N$ is said to be in arithmetic progression if $$S=\{a,a+d,a+2d,\ldots\}$$ where $a,d\in\Bbb N$. Here $d$ is called the step of $S$. We are asked to show that $\Bbb N$ cannot be partitioned into a finite number of subsets that are in arithmetic progression with distinct steps (except for the case $a=d=1$). 

He gives a hint to write $$\sum_{n\in\Bbb N}z^n$$ as a sum of terms of the type $$\frac{z^a}{1-z^d}.$$ How do I apply the hint? I know that $$\sum_{n\in\Bbb N}z^n=\frac{z}{1-z}$$ but that doesn't have anything to do with the $a$ or $d$. Thanks for any help!

Suppose $\mathbf{N}=S_1\cup\cdots S_k$ is a partition of $\bf N$ and $S_r=\{a_r+d_rm:m\ge0\}$. Then
$$\begin{array}{cl}\frac{z}{1-z} & =\sum_{n\in\bf N}z^n \\ 
& =\sum_{r=1}^k\left(\sum_{n\in S_r}z^n\right) \\
& = \sum_{r=1}^k\left(\sum_{m\ge0}z^{a_r+d_rm}\right) \\
& = \sum_{r=1}^k\frac{z^{a_r}}{1-z^{d_r}}.\end{array}\tag{$\circ$}$$
Suppose each $d_r$ is distinct. Do both sides of $(\circ)$ have the same poles in the complex plane?
