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If a/b = b/c = c/d = d/e, prove that (ab + bc + cd + de)² = (a² + b² + c² + d²)(b² + c² + d² + e²). Which statement correctly characterizes this identity?

1. It is true and follows from the equality condition of the Cauchy-Schwarz inequality (terms proportional)
2. It is true only when a = b = c = d = e
3. It is false in general
4. It holds only for positive a, b, c, d, e

Correct answer: It is true and follows from the equality condition of the Cauchy-Schwarz inequality (terms proportional)

Solution

With common ratio r: a, b=ar, c=ar², d=ar³, e=ar⁴. LHS terms ab+bc+cd+de = a² r(1 + r² + r⁴ + r⁶). a²+b²+c²+d² = a²(1+r²+r⁴+r⁶). b²+c²+d²+e² = a² r²(1+r²+r⁴+r⁶). Product of the two = a⁴ r² (1+r²+r⁴+r⁶)² = (LHS)². Since (a,b,c,d) is proportional to (b,c,d,e), Cauchy-Schwarz holds with equality, confirming the identity.

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