The sum of the three numbers that make up a geometric progression is 3, and the sum

The sum of the three numbers that make up a geometric progression is 3, and the sum of their squares is 21. Find these numbers.

1. For the geometric progression B (3) it is known:

2. The sum of the three terms (numbers) is equal to:
S3 = B1 + B2 + B3 = B1 + B1 * q + B1 * q² = B1 * (1 + q + q²) = 3;

3. Let’s calculate right away (I won’t write everything down):
S3² = (B1 * (1 + q + q²) ² = 3² =
B1² * (1 + q + q ^ 4) + 2 * q * B1² * (1 + q + q²) = 9;

4. Sum of squares of numbers:
Sk = B1² + B2² + B3² = B1² + (B1 * q) ² + (B1 * q²) ² = B1² * (1 + q² + q ^ 4) = 21;

5. Subtract:
S3² – Sk = (B1² * (1 + q + q ^ 4) + 2 * q * B1² * (1 + q + q²)) – (B1² * (1 + q² + q ^ 4)) = 9 – 21;
2 * q * B1² * (1 + q + q²) = -12;
(B1 * q) * (B1 * (1 + q + q²) = -6;
B2 * S3 = -6;
B2 = (-6) / S3 = (-6) / 3 = -2;

6. Let’s transform the sum of three numbers:
S3 = B1 + B2 + B3 = B1 + (-2) + B1 * q² = 3;
B1 * (1 + q²) = 3 + 2 = 5;
(B1 * q) * (1 + q²) = 5 * q;
(-2) * (1 + q²) = 5 * q;

7. Solve the equation:
2 * q² + 5 * q + 2 = 0;
q1,2 = (-5 + – sqrt ((- 5) ² – 4 * 2 * 2) / (2 * 2) = (-5 + – 3) / 4;

8.q1 = (-5 + 3) / 4 = -0.5;
B1 = B2 / q = (-2) / (-0.5) = 4;
B2 = -2;
B3 = B2 * q = (-2) * (-0.5) = 1;

9.q2 = (-5 – 3) / 4 = -2;
B1 = B2 / q = (-2) / (-2) = 1;
B2 = -2;
B3 = B2 * q = (-2) * (-2) = 4; (mirrors)
Answer: numbers 1, -2, 4.