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Decline of Roman Empire

If a, b, c, d...

Question

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 show that a, b, c and d are in G.P.

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Solution

$(a^{2} + b^{2} + c^{2}) p^{2} - 2 (a b + b c + c d) p + (b^{2} + c^{2} + d^{2}) \leq 0 \Rightarrow (a^{2} p^{2} + b^{2} p^{2} + c^{2} p^{2}) - 2 (a b p + b c p + c d p) + (b^{2} + c^{2} + d^{2}) \leq 0 \Rightarrow (a^{2} p^{2} - 2 a b p + b^{2}) + (b^{2} p^{2} - 2 b c p + c^{2}) + (c^{2} p^{2} - 2 c d p + d^{2}) \leq 0 \Rightarrow {(a p - b)}^{2} + {(b p - c)}^{2} + {(c p - d)}^{2} \leq 0 \Rightarrow {(a p - b)}^{2} + {(b p - c)}^{2} + {(c p - d)}^{2} = 0 \Rightarrow {(a p - b)}^{2} = 0 \Rightarrow p = \frac{b}{a} Also, {(b p - c)}^{2} = 0 \Rightarrow p = \frac{c}{b} Similiarly, \Rightarrow {(c p - d)}^{2} = 0 \Rightarrow p = \frac{d}{c} ∴ \frac{b}{a} = \frac{c}{b} = \frac{d}{c} Thus, a, b, c and d are in G . P .$

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