A geometric series has 3 terms which add up to 42. What are the terms if the third term is 3.2 times the sum of the first two terms?

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Consecutive terms of a GP have a common ratio. Let the terms we have to find be a, ar and ar^2.

The third term is 3.2 times the sum of the first two

=> ar^2 = 3.2(a + ar)

=> r^2 - 3.2r - 3.2 = 0

=> r^2 - 4r + 0.8r - 3.2 = 0

=> r(r - 4) + 0.8(r - 4) = 0

=> (r + 0.8)(r - 4) = 0

=> r = -0.8 and r = 4

The sum of the three numbers is 42

For r = -.08

a( 1 - 0.8 + .64) = 42

=> a = 50

For r = 4

a(1 + 4 + 16) = 42

=> a = 42 / 21

=> a = 2

Using the values of a and r that we have found the 3 numbers are 50 , -40 , 32   or 2 , 8 , 32

The required numbers are (50, -40, 32) and (2 , 8, 32)

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