Call the 5 numbers x, x + y, x + 2y, x + 3y, x + 4y, where x is the value of the first number and y is the constant value difference.
Since their sum is 20, 5x + 10y = 20.
Since the product of the first and last number is 15, x^2 + 4xy = 15.
From 5x + 10y = 20, we can see that x + 2y = 4.
-> (x+2y)^2 = 16 -> x^2 + 4xy + 4y^2 = 16 -> 4y^2 = 1.
-> y = 1/2 or -1/2
Replace back into the original equation : 5x + 10.(1/2) = 20 or 5x + 10.(-1/2) = 20
-> 5x + 5 = 20 / 5x + (-5) = 20
-> x = 3 / x = 5.
From the 4 combinations of x and y, we see that (x;y) = (5; -1/2) and (3; 1/;2) would satisfy the second condition.
So, the five numbers that we need to find has 2 different combinations :
(5; 9/2; 4; 7/2; 3) or vice versa (sum = 20)