# Find the value of r in the following case.The numbers x, y and z are the first three terms if a geometric sequence with common ratio r, and also the first, second and fourth terms of an arithmetic...

Find the value of r in the following case.

The numbers x, y and z are the first three terms if a geometric sequence with common ratio r, and also the first, second and fourth terms of an arithmetic sequence.

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### 1 Answer

The numbers x, y and z are the first three terms if a geometric sequence with common ratio r, and they are also the first, second and fourth terms of an arithmetic sequence.

Now the terms of a GP are written as a*r^(n-1)

So x = a*r^0

y = a*r

z = a*r^2

Now they are also the 1st , 2nd and 4th terms of an AP. So the difference between the 4th and the 2nd terms is twice that between the 2nd and 1st terms.

So 2*(y - x) = ( z - y)

=> 2(ar - a) = ( ar^2 - ar)

=> 2a ( r - 1) = ar(r - 1)

=> 2 = r

=> r = 2

**Therefore the common ratio is 2.**