If x, y, z are in AP and x^2, y^2 and z^2 are in HP then prove that its possible - Maths - Sequences and Series

Question

If x, y, z are in AP and x^2, y^2 and z^2 are in HP then prove that
its possible when x, y, z are equal or when -x/2, y, z are in GP

Ishwarmani · Accepted Answer

Dear Student,Please find below the solution to the asked query:∵x,y,z are in AP∴ y = x + z2     
(1)and x2,y2,z2 are in HP∴ y2 = 2x2z2x2+z2      
(2)from (1) and (2),⇒x + z22 = 2x2z2x2+z2⇒x + z2x2+z2 = 8x2z2⇒x2+z2+2xzx2+z2 = 8x2z2⇒x2+z22+2xzx2+z2-8x2z2 = 0⇒x2+z22-4x2z2 +2xzx2+z2-4x2z2 = 0⇒x2-z22+2xzx2+z2-2xz = 0⇒x2-z22+2xzx-z2 = 0⇒x-z2x + z2 + 2xz = 0when x-z2          x = zand from 1,y = x + x2 = x∴x = y = zwhen x + z2 + 2xz = 0⇒x2+z2+4xz=0⇒x2+z2=-4xzand from 2,putting   x2+z2 = 2x2z2y2⇒2x2z2 y2  =-4xz⇒xz = -2y2⇒y2 = -x2× z∴-x2,y,z are in GP
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If x, y, z are in AP and x^2, y^2 and z^2 are in HP then prove that its possible when x, y, z are equal or when -x/2, y, z are in GP