Question

If a, b, c are three consecutive terms of an ap and x, y, z are three consecutive terms of g.p then prove that x^b-c, y^c-a, z^a-b = 1​

Answer 1

If a, b, c are three consecutive terms of an ap and x, y, z are three consecutive terms of g.p then prove that x^b-c, y^c-a, z^a-b = 1​

• Given,
• a, b, c are three consecutive terms of an ap
• x, y, z are three consecutive terms of g.p
• let, d be the common difference between the terms of ap.
• So, we have,
• a = a, b = a+d, c = a+2d
• Given, x^b-c, y^c-a, z^a-b
• = x^{ a+d - (a+2d)} * y^{a+2d - a} * z^{a - (a+d)}
• = x^{-d} * y^{2d} * z^{-d}
• = (xz)^{-d} * (y^2)^{d}
• since, x, y and z are in gp, we have
• y/x = z/y
• ⇒ y^2 = xz
• now, we have,
• = (y^2)^{-d} * (y^2)^{d}
• = (y^2)^{-d+d}
• = (y^2)^{0}
• = 1
• Therefore, x^b-c . y^c-a . z^a-b =1