Can a geometric sequence have common ratio of $1$? (eg: $2,2,2,2,2,...$)
it follows the rule $b^2 = ac$. So, I think it is also a geometric progression.
But the sum formula is $\frac{a(r^{n}-1)}{(r-1)}$ and when $r=1$ , denominator gets zero. Hence, I got this doubt.
I know the formula is when $r>1$ or $r<1$. but why it is not defined for $r=1$. Is it because when $r=1$, it is not a GP?
Please help. thanks
A geometric sequence $u_n$ is a sequence which is defined recursively by (given $u_1$)
$$u_{n+1}=ru_n,\quad r\in \Bbb R $$ where the ratio $r$ is a given number. The sum formula is $$\sum_{k=1}^nu_k=u_1\frac{1-r^n}{1-r},\;\text{if}\, r\ne1$$ and $$\sum_{k=1}^nu_k=nu_1,\;\text{if}\, r=1$$