SOLUTION: a^(1/x)=b^(1/y)=c^(1/z) if a,b,c are in GP. Prove x,y,z are in AP

Question 779653: a^(1/x)=b^(1/y)=c^(1/z) if a,b,c are in GP. Prove x,y,z are in AP
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
One way to solve this is...

If a, b and c are in a geometric progression (which I assume "GP" means) then consecutive terms have a common (fixed) ratio. If we call this ratio "r" then

or b = a*r
and

or c = b*r

Since b = a*r, we can write c in terms of a:

Substituting these expressions in "a" for "b" and "c" into the given equation we get:

To see if x, y and z are in an arithmetic progression (AP), where consecutive terms have a common (fixed) difference, we will start by expressing y and z in terms of x. First we'll do y:

First let's eliminate the fractions in the exponents. Raising both sides to the LCD power:

which simplifies to:

Now we use logarithms to get the x's and y's out of the exponents. Finding the base a log of each side:

Using a property of logs, the exponents in the arguments can be moved out in front:

The log on the left is just a 1. On the right we can use another property of logs to split it into two logs (separating the "a" and the "r"):

The first log on the right is a 1 so this simplifies to:

Now we repeat the process for z:

If x, y and z are in an AP then they should have a common difference (which we will call "d"). Let's see:

Substituting in for y:

The x's cancel:

Now lets try

Substituting for both z and y:

Again the x's cancel:

These are like terms so we can subtract them:

As we can see, the two differences are the same. So x, y and z are in an AP.
RELATED QUESTIONS
if a,b,c in ap nd x,y,z in gp prove that x^b y^c z^a=x^c y^a... (answered by fractalier)

If 1/y+ax,1/2y,1/y+az are in AP prove that a^2 x,y,z are in... (answered by richwmiller)

If a,b,c, are in G.P and a^x=b^y=c^z the x,y,z are in 1)H.P 2)G.P 3)A.P (answered by Edwin McCravy)

If a^2,b^2,c^2 are in AP then prove that 1/b+c,1/c+a,1/a+b are also in... (answered by Edwin McCravy)

if a,b,c are in an arithmetic progression and x,y,z are in a geometric progression prove... (answered by ikleyn)

If a,b,c are in an arithmetic progression and x,y,z are in a geometric progression, prove (answered by ikleyn)

if x=c*y+b*z y=a*z+c*x z=b*x+a*y Then prove that x^2/1-a^2=y^2/1-b^2=z^2/1-c^2 (answered by Edwin McCravy)

If a^x=bc;b^y=ca;c^z=ab then prove x/(x+1) + y/(y+1) + z/(z+1)... (answered by Edwin McCravy)

a^x =b^y = c^z = abc ; find 1/x + 1/y + 1/z (answered by tommyt3rd)