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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 $10$.

harisri
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    Hint: Remember that the $n$:th term of a geometric series is of the form $a_0 q^{n-1}$, where $a_0$ is the first term and $q$ is a constant. Therefore, the tenth term is $a_0 q^9$. When you know $a_0$ (the first term) and the ninth terms, how would you proceed? – Matti P. Jul 10 '19 at 10:26
  • The ratio of the tenth term to the first term is the ninth power of the common ratio. So to find the common ratio, divide the tenth term by the first term and then take the ninth root of the quotient. – bof Jul 10 '19 at 10:27
  • Please, use MathJax – nonuser Jul 10 '19 at 10:45
  • The linked question describes how to find the common ratio of a sequence where the first and fourth term are known. There's no real difference between that and the question you're asking. So your added parentheses is just wrong. – Henrik supports the community Jul 11 '19 at 06:26

2 Answers2

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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.

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

Dr. Sonnhard Graubner
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