Show that the common ratio between any two consecutive terms of a geometric sequence is a constant r.

Question

I think I have to write some kind of proof for the question, but I am not sure how to actually write it. In some of my previous questions I used examples to write the proofs, but my lecturer marked them wrong because he wanted formal proofs. The question is: Show that the common ratio between any two consecutive terms of a geometric sequence is a constant r.

Answer 1

Let the first term be $a$. Then, since this is a geometric sequence, we have the sequence will look like this $$a, ar, ar^2,...,ar^k,....$$ Of course, it follows that the common ratio of this sequence is $$\frac{ar^m}{ar^{m-1}}=r.$$