Is it always possible to transform one arithmetic progression into another?

Question

Suppose we are given two arithmetic progressions $a, a+h, a+2h, ...$ and $c, c+l, c+2l, ...$ Is it always possible to find a linear function $y=kx+b$ which transforms the first progression into the second?

From "Functions and Graphs", Gelfand

In the book we learned that a linear function converts one arithmetic progression into another. And examples involvig numbers also make sense for me.

However, I can't figure out how to approach the above problem. How do we show if it's possible to tranform one arithmetic progression into another?

Answer 1

Hint: Try to find $k$ and $b$ such that $a$ maps to $c$ and $a+h$ maps to $c+l$. Then check what $a+nh$ gets mapped to. You may notice that only one property of $h$ may lead to failure.

Answer 2

The answer is yes.

Try $$ f(x)=c+(l/h)(x-a)$$, then $$f(a)=c, f(a+h)=c+l, f(a+2h)=c+2l,...$$