if a,b,c,d and p are different real numbers such that
(a2+ b2+c2)p2 -2(ab+bc+cd) p +(b2+c2+d2) ≤ 0 then
a,b,c and d are in geometric progression
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1 Answers
2305By AM GM inequality we have,
(ap)2 + b2 ≥ 2abp
(bp)2 + c2 ≥ 2cbp
(cp)2 + d2 ≥ 2cpd
Therefore,
(ap)2 + b2+ (bp)2 + c2 +(cp)2 + d2 - 2cpd - 2cbp - 2abp ≥ 0
Since the above condition is given as RHS≤0
(ap)2 + b2+ (bp)2 + c2 +(cp)2 + d2 - 2cpd - 2cbp - 2abp = 0
Equality holds when all the 3 inequalities are equal.
This is only possible when ap = b, bp = c and cp = d
Or,ba = cb = cd=p
Hence they are in GP.
