Question: 1 of 3 terms of GP is -24, if their sum is -56/3 and their product is 512, find the other 2.
Let the three terms of the GP be: a, ar and ar^2, where r is the common ratio.
Sum is -56/3, i,.e. a(1 + r + r^2) = -56/3
Product is 512, i.e. a*ar*ar^2 = 512
i.e. (ar)^3 = 512 = 8^3
or, ar = 8
We are told, 1 of 3 terms is -24, i.e.
a = -24 implies r = -1/3 (since ar = 8)
or,
ar = -24 (this option not valid since we are given that ar = 8)
or,
ar^2 = -24 implies r = -3 since ar = 8
but ar^2 = -24 with r = -3 implies a = -24/9 = -8/3
and ar = 8 with r = -3 implies a = -8/3
This leaves us with the 1st option, a = -24 with r = -1/3 and the 3rd option, a = -8/3 with r = -3
Check
The three elements are: -24, 8, - 8/3, OR -8/3, 8, -24 (i.e. a, ar, ar^2)
It should be noted that both options give the same three elements, just in different orders.
Sum = -24 + 8 - 8/3 = -16 - 8/3 = -48/3 - 8/3 = -56/3 -- check
Product = (-24)*(8)*(-8/3) = (-8)*(8)*(-8) = 512 -- check
Answer: The three elements are: -24, 8, -8/3