Question

if x^(p)=y^(q)=z^(r)and (y)/(x)=(z)/(y) then prove that (2)/(q)=(1)/(p)+(1)/(r)

Answer 1

Solution:

Given : x^p = y^q = z^r

y/x = z/y

To prove : 2/q = 1/p + 1/r

Proof :

We have ;

x^p = y^q = z^r

Let ,

x^p = y^q = z^r = k

If x^p = k

then x = k^(1/p) -------(1)

If y^q = k

then y = k^(1/q) -------(2)

If z^r = k

then z = k^(1/r) --------(3)

Also,

It is given that ;

y/x = z/y

Thus,

=> y/x = z/y

=> y•y = z•x

=> y² = xz

Using eq-(1) , (2) and (3) , we have ;

=> [ k^(1/q) ]² = [ k^(1/p) ]•[ k^(1/r) ]

=> k^(2/q) = k^(1/p + 1/r)

=> 2/q = 1/p + 1/r

Hence proved .