Question

Pls Help :(((

If a^x=b^y=c^z is a system of exponential expression where b^2=ac


Prove that 1/x + 1/z = 2/y

Answer 1

`\text{Let,}\\\\~~~~~~~a^x =b^y = c^z = k\\\\\text{So,}\\ \\a = k^(\frac 1x)\\\\b=k^(\tfrac 1y)\\\\\ c = k^(\tfrac1z)\\\\\text{Now, given that,}\\\\~~~~~~~~~b^2 = ac\\\\\implies \left( k^\tfrac 1y \right)^2 = k^(\tfrac 1x) \cdot k^(\tfrac 1z)\\\\\implies k^(\tfrac 2y) = k^(\tfrac 1x + \tfrac 1z) ~~~~~~~~~~;[a^m \cdot a^n = a^(m+n)]\\\\\implies \frac 2y = \frac 1x + \frac 1z\\\\\implies \frac 1x + \frac 1z = \frac 2y \\\\\text{Proved.}`