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What are the different kinds of relations one can obtain between the elements of a finite Geometric progression other than the fact that one element upon the previous element gives the common ratio?

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  • $\begingroup$ Another common one can be written as $$a_r=\sqrt{a_{r-1}a_{r+1}}.$$ But you have to note that "all are equivalent to the definition. $\endgroup$ – Bumblebee Jan 11 '15 at 13:16
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For example: if $\;a_1,a_2,...,a_n\;$ is a G.P., then there exists constant $\;q\;$ such that

$$\begin{cases}\frac{a_{k+1}}{a_k}=q&,\;\;1\le k<n\\{}\\a_k^2=a_{k-1}a_{k+1}&,\;\;2\le k<n\\{}\\a_k=a_1q^{k-1}&,\;\;1\le k\le n\\{}\\a_1+a_2+\ldots+a_n=a_1\frac{q^n-1}{q-1}&,\;\;q\neq1\end{cases}$$

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The logarithms follow an arithmetic progression.

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