Let's call the three positive numbers in the arithmetic progression "a", "b", and "c".
The sum of the squares of the numbers is 155, so:
a^2 + b^2 + c^2 = 155
The sum of the numbers is 21, so:
a + b + c = 21
We can use the second equation to solve for one of the numbers in terms of the others. Let's solve for "a":
a = 21 - b - c
Now we can substitute this expression for "a" into the first equation:
(21 - b - c)^2 + b^2 + c^2 = 155
Expanding the square:
441 - 42b + b^2 - 42c + 2bc + c^2 = 155
Rearranging the equation:
b^2 - 42b + c^2 - 42c = 286
This is a quadratic equation in two variables, b and c. To find the values of b and c, we can either use the quadratic formula or try to factor the left-hand side. Since the equation is symmetric in b and c, we can try to factor it into a difference of squares:
(b - 21)^2 - (c - 21)^2 = 286
Expanding the squares:
b^2 - 42b + 441 - c^2 + 42c - 441 = 286
Rearranging the equation:
b^2 - 42b + c^2 - 42c + 441 - 441 = 286 + 441 - 441
Simplifying the right-hand side:
b^2 - 42b + c^2 - 42c + 441 = 727
Factorizing the left-hand side:
(b - 21)^2 - (c - 21)^2 = 727
Taking the square root of both sides:
|b - 21| - |c - 21| = 27
Since both b and c are positive numbers, we know that the value of the difference on the right-hand side is positive. Therefore, we can subtract 21 from both sides to get:
b - c = 48
So, b = c + 48.
We can now use the equation a = 21 - b - c to find a:
a = 21 - (c + 48) - c
Simplifying:
a = 21 - 2c - 48
a = -2c + -27
So, the numbers are -2c - 27, c + 48, and c