Question

if x, y, z are positive real numbers and p, q, r are natural numbers such that x^p=y^q=z^r and y/x=z/y, then prove that 2/q=1/p+1/r​

Answer 1

Answer:

23

Step-by-step explanation:

Answer 2

Let us suppose that the value of powers are $k$:-

$\rightarrow x^p=k,\ y^q=k,\ z^r=k$

Then

$\rightarrow x=k^{\frac{1}{p}},\ y=k^{\frac{1}{q}},\ z=k^{\frac{1}{r}}\ \cdots(1)$

It is given that:-

$\rightarrow \dfrac{y}{x}=\dfrac{z}{y}$

Thus

$\rightarrow y^2=zx$

By substituting $(1)$:-

$\rightarrow(k^{\frac{1}{q}})^2=k^{\frac{1}{p}}k^{\frac{1}{r}}$

$\rightarrow k^{\frac{2}{q}}=k^{\frac{1}{p}+\frac{1}{r}}\ \therefore\dfrac{2}{q}=\dfrac{1}{p}+\dfrac{1}{r}$

Hence proven.