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.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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The numbers 2^x, 2^y and 2^z are the terms of a geometric progression. We have to determine x, y and z which are the terms of an arithmetic progression.

As the consecutive terms of a geometric progression have a common ratio : 2^y/2^x = 2^z/2^y

=> 2^(y - x) = 2^(z - y)

=> y - x = z - y

=> 2y = x + z

This is true for all values of x, y and z that are consecutive terms of an arithmetic progression.

x, y and z do not have a unique value, they can be any three consecutive terms of any arithmetic progression.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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).

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