MHB Find Lowest Value for A: a1, a2, a3 & 4 | Arithmetic Progression

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The discussion focuses on finding the lowest value of A = a1a2 + a2a3 + a3a1 for an arithmetic progression defined by a1, a2, a3, and 4. A participant expresses uncertainty about their approach, having derived A = 3x^2 + 6xd + 2d^2. Another user suggests that the fourth term of the progression can simplify A into a single variable, which would facilitate finding the minimum value. The original poster acknowledges the oversight and appreciates the guidance provided. The conversation emphasizes the importance of utilizing all given terms in mathematical problems for simplification.
mitaka90
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a1, a2, a3 and 4 make an arithmetic progression with difference d. For which values of d, A = a1a2 + a2a3 + a3a1 has the lowest value?I don't know if I went with the right approach, but I managed to get this : A=3x2 +6xd + 2d2 for a1= x, a2 = x + d, etc... But I don't know what else to do.
 
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mitaka90 said:
a1, a2, a3 and 4 make an arithmetic progression with difference d. For which values of d, A = a1a2 + a2a3 + a3a1 has the lowest value?I don't know if I went with the right approach, but I managed to get this : A=3x2 +6xd + 2d2 for a1= x, a2 = x + d, etc... But I don't know what else to do.
Hi mitaka90!

It seems to me you haven't made good use of the given fourth term in that arithmetic progression...:) the fourth term would help you to simplify your $A$ in terms of only one variable and when you have the quadratic equation in terms of one variable, I believe you could handle from there...
 
anemone said:
Hi mitaka90!

It seems to me you haven't made good use of the given fourth term in that arithmetic progression...:) the fourth term would help you to simplify your $A$ in terms of only one variable and when you have the quadratic equation in terms of one variable, I believe you could handle from there...

Omg, I'm such a moron. I hate it when I do the hard work and then the easiest and most noticable thing just slips from my sight. Thank you sincerely, I guess that little tip is what I needed.
 

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