In a geometric sequence, only the first term and the tenth term are given. What is the correct method to find the common ratio using this data?

Question

In this geometric sequence for an exponential model I will be graphing based on this data,

term $1 = 50,000$ and term $10 = 309,587$.

I do not know how to find the common ratio based solely on this data. (This is not the same as "how to find the common ratio of a geometric sequence) because my question is on, how would you find the common ratio with a missing internal number, say, term 1 and term 10 as opposed to term 1 and term 2.

I know that if this were an arithmetic sequence, I would convert all of my data into an explicit equation $f(n)=f(1)+d(n-1)$ in this case,

$$ \begin{align*} 309,587 &= 50,000 + (10-1)d\\ \implies 309587 - 50000 &= 9d\\ \implies d &= 28843 \end{align*} $$

I do not think this would be the same for a geometric sequence for example $f(n) = f(1) * d(n-1)$ or in this case, $309,587 = 50,000 * (10-1)d$ and then do $309587 - 50000 = 9d$, $d = 28843$ as for the arithmetic sequence, because that is arithmetic(adding) and this is geometric (multiplying).

I need to know how to find the common ratio of a geometric sequence given term $1$ and term $10$.

Answer 1

If the first term is $a$ and the common ration is $r$, then the $10$^th term is $a\times r^9$ and therefore $r$ is the ninth root of the quotient between the $10$^th term and the first one.

Answer 2

We are using the formula $$a_n=a_1q^{n-1}$$ so $$q=\sqrt[9]{\frac{309.587}{50000}}$$