Geometric Sequence with common ratio $1$

Question

Does $2,2,2,\dots$ form a geometric sequence?

According to our professor, it is not considered a geometric sequence. But it follows the rule with $r=1$.

Can anyone please explain to me why or why not is it a geometric sequence? Thanks.

Answer 1

You can, see Wikipedia for example.

But it makes a mess of the sum formula:

$$\sum_{i=0}^n ar^i = a\frac{1-r^{n+1}}{1-r}$$

Answer 2

A sequence $(a_n)_{n \ge 0}$ is geometric iff there is $r$ such that $a_n=a_0r^n$ for all $ n \ge 0$.

In your case we have: $a_0=2$. Then $2=a_1=2*r$, hence $r=1$. With this $r$ we indeed have

$2=a_n=a_0r^n$ for all $ n \ge 0$.

Consequence: $(2,2,2,...)$ is geometric and your professor is wrong.