Saturday, February 20, 2016

What are the integer numbers x,y,z if they are the terms of an artihmetic progression.2^x,2^y, 2^z are the terms of geometric progression.

If x,y,z are the terms of an AP, we can apply the mean
value theorem:


y =
(x+z)/2


We'll cross multiply and we'll
get:


2y = (x+z) (1)


If
2^x,2^y,2^z are the terms of a GP then 2^y is the geometric mean of 2^x and
2^z:


2^y = sqrt(2^x*2^z)


We'll
square raise both sides:


2^2y =
2^x*2^z


Since the bases of the exponentials from the right
side are matching, we'll add the exponents:


2^2y =
2^(x+z)


Since the bases are matching, we'll apply one to
one property:


2y =
(x+z)


Therefore, the integer terms x,y,and z
may have any value, as long as they respect the constraint 2y =
(x+z).

No comments:

Post a Comment

What accomplishments did Bill Clinton have as president?

Of course, Bill Clinton's presidency will be most clearly remembered for the fact that he was only the second president ever...