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Arithmetic Progression Problem

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anemone

MHB POTW Director
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Feb 14, 2012
3,690
Find three irreducible fractions $\dfrac{a}{d}$, $\dfrac{b}{d}$ and $\dfrac{c}{d}$ that form an arithmetic progression, if $\dfrac{b}{a}=\dfrac{1+a}{1+d}$, $\dfrac{c}{b}=\dfrac{1+b}{1+d}$.
 

mente oscura

Well-known member
Nov 29, 2013
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Find three irreducible fractions $\dfrac{a}{d}$, $\dfrac{b}{d}$ and $\dfrac{c}{d}$ that form an arithmetic progression, if $\dfrac{b}{a}=\dfrac{1+a}{1+d}$, $\dfrac{c}{b}=\dfrac{1+b}{1+d}$.
Hello.

[tex]If \ a<d \rightarrow{}a>b>c[/tex]

[tex]a<d \rightarrow a+1<d+1[/tex]

Same "c".

[tex]k=a-b, k=[/tex]reason of the arithmetic progression.

[tex]k=a-\dfrac{a^2+a}{d+1}=\dfrac{ad-a^2}{d+1}[/tex]

[tex]b-k=c \rightarrow{}b-\dfrac{ad-a^2}{d+1}=\dfrac{b^2+b}{d+1}[/tex]

Resolving:

[tex]a^2-ad-b^2+bd=0[/tex]

[tex]a=\dfrac{d \pm \sqrt{d^2+4b^2-4bd}}{2}[/tex]

[tex]a=\dfrac{d \pm \sqrt{(d-2b)^2}}{2}[/tex]

[tex]a=d-b \ or \ a=b[/tex]

[tex]a=d-b \rightarrow{}d=a+b[/tex]

[tex]\dfrac{b}{a}=\dfrac{1+a}{1+a+b} \rightarrow{}a^2+a=b^2+b+ab[/tex]

[tex]a=\dfrac{(b-1) \pm \sqrt{(b-1)^2+4b^2+4b}}{2}[/tex]

[tex](b-1)^2+4b^2+4b=5b^2+2b+1=T^2[/tex]

[tex]For \ b=2, T=5[/tex]

[tex]a=3, b=2, c=1, d=5[/tex]



Regards.
 
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anemone

MHB POTW Director
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Feb 14, 2012
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Thanks for participating, mente oscura! Your answer is correct, bravo!