If a, b, c, d and p are different real numbers such that (a 2 + b 2 + c 2 ) p 2 – 2 (ab + bc + cd) p + (b 2 + c 2 + d 2 ) ≤ 0, then a, b, c, d are in -
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We have
(a 2 + b 2 + c 2 ) p 2 – 2 (ab + bc + cd) p + (b 2 + c 2 + d 2 ) ≤ 0 …(i)
LHS = (a 2 p 2 – 2abp + b 2 ) + (b 2 p 2 – 2bcp + c 2 ) +
(c 2 p 2 – 2cdp + d 2 )
= (ap – b) 2 + (bp – c) 2 + (cp – d) 2 ≥ 0 …(ii)
Since, the sum of squares of real number is non-negative
From Equations (i) and (ii)
⇒ (ap – b) 2 + (bp – c) 2 + (cp – d) 2 = 0
⇒ ap – b = 0 = bp – c = cp – d
⇒ 
∴ a, b, c, d are in G.P.
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