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If a,b,c are in arithmetic progression., p,q,r in harmonic progression and ap,bq,cr are in geometric progression., then $\frac{p}{r}+\frac{r}{p} = $ ?

EDIT: I have tried to use the basic/standard properties of the respective progressions to get the desired result, but I am not yet successful.

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3 Answers 3

Let's represent your arithmetic progression as


where $h=b-a$

and your harmonic progression as


(following Wikipedia)

and the condition that the products of corresponding terms of the arithmetic and harmonic sequences form a geometric sequence implies




we solve for $l$, which has two solutions.

I leave the rest to you. ;)

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If a,b,c are in arithmetic progression, then you can write them as a, a+d, a+2d. Maybe all you need is 2b=a+c. Similarly for geometric progression, you can write them as ap, apz, apz^2 where z is the common ration (often we use r, but that was taken). Again, maybe (bq)^2=ap*cr is all you need.

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I knew this relations and I tried to use them but not quite successful. –  Damir Frezivioski Oct 27 '10 at 15:54

Notice that $\rm\:\ \quad\displaystyle \frac{p}r+\frac{r}p\ =\ \frac{(p+r)^2}{pr} - 2$

But we have that$\rm\quad\displaystyle p\:r\ =\ \frac{(bq)^2}{ac}\ \ $ via $\rm\ ap,\:bq,\:cr\ $ geometric

and we have $\rm\quad\ \ \displaystyle p+r\ =\ \frac{2pr}q\quad\ \ \ $ via $\rm\ p,q,r\ $ harmonic

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