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If $p,q$, and $r$ are terms of an arithmetic progression are also in a geometric progression, then find the common ratio of the geometric progression in terms of $p,q$, and $r$.

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I considered First term as 'a' and common difference as 'd'. And Pth term as a + (p-1)d .And similarly q and r. –  vikiiii Feb 24 '12 at 4:27
    
And then i did q^2=p * r. But i dont know how to eliminate a and d.Since there is only one equation. –  vikiiii Feb 24 '12 at 4:28
    
I also do not know if (sequences-and-series) is the proper tag, but I think it's a better fit than (calculus). If anyone has a better idea for a tag, feel free to change it. –  JavaMan Feb 24 '12 at 4:28
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1 Answer

up vote 2 down vote accepted

Hint: Let $p,q,r$ be the terms. Let $x$ be the ratio in the geometric progression, $y$ be the increment in the arithmetic progression. $x=\frac qp, y=q-p$ Then the terms are $p, px, px^2$ and also $p, p+y, p+2y$. Can find two equations for $x$ and $y$?

Added: You have $p+y=px, p+2y=px^2$. We have two equations in two unknowns if we regard $p$ as a parameter. Subtracting, $y=px^2-px$. Dividing, $x=\frac rq=\frac {p+2y}{p+y}$. I suspect the desired answer is the solution for $x$ of these two.

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Doesn't this assume that $p,q$, and $r$ are adjacent terms in each progression? –  JavaMan Feb 24 '12 at 4:31
    
Ya..i think the same..they are not necessarily adjacent terms. –  vikiiii Feb 24 '12 at 4:34
    
@JavaMan: Absolutely. But without that, I don't think there is an answer. –  Ross Millikan Feb 24 '12 at 4:36
    
@anna: see above-I can't ping two people with one comment. Hope this helps. –  Ross Millikan Feb 24 '12 at 4:38
    
@RossMillikan what i tried i have put in the comment.Please let me know if i am approaching correct. –  vikiiii Feb 24 '12 at 4:42
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