# If the 2nd, 4th and 5th terms of an arithmetic sequence are the first three terms of a geometric sequence what can be common ratio of the geometric sequence.

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justaguide | Certified Educator

The nth term of an arithmetic series is given by a + (n - 1)d and the nth term of a geometric series is a'r^(n-1)

The second term of the arithmetic series is the 1st term of the geometric series:

=> a + d = a' ...(1)

The fourth term of the arithmetic series is the 2nd term of the geometric series:

=> a + 3d = a'r ...(2)

The fifth term of the arithmetic series is the 3rd term of the geometric series:

=> a + 4d = a'r^2 ...(3)

(2) - (1)

=> 2d = a'(r - 1)

(3) - (2)

=> d = a'(r^2 - r)

Dividing the two

`(r^2 - r)/(r - 1) = 1/2`

=> 2r^2 - 2r = r - 1

=> 2r^2 - 3r + 1 = 0

=> 2r^2 - 2r - r + 1 = 0

=> 2r(r - 1) - 1(r - 1) = 0

=> (2r - 1)(r - 1) = 0

=> r = `1/2` and r = 1

**The common ratio of the geometric series can be `1/2` and 1**