let x be AM and y ,z be two GM between any two positive number prove that y3 + - Maths - Sequences and Series

Question

let x be AM and y ,z be two GM between any two positive number
prove that y3 + z3 = 2xyz

Aakash Sharma · Accepted Answer

Let a and b be two positive numbers Then x = A M of a and b = $\frac{a + b}{2}$ and a, y, z, b will be in G P with $common ratio (r) = (\frac{b}{a})^{\frac{1}{2 + 1}} = (\frac{b}{a})^{\frac{1}{3}} \Rightarrow y = a r = a (\frac{b}{a})^{\frac{1}{3}} = a^{\frac{2}{3}} b^{\frac{1}{3}} and z = a r^{2} = a^{\frac{1}{3}} b^{\frac{2}{3}} ∴ y^{3} + z^{3} = a^{2} b + a b^{2} = a b (a + b) = 2 a b x and y z = a b \Rightarrow y^{3} + z^{3} = 2 x y z$

let x be AM and y ,z be two GM between any two positive number. prove that y3 + z3 = 2xyz.

let x be AM and y ,z be two GM between any two positive number. prove that y³+ z³ = 2xyz.