The difference between the first and second terms of the geometric progression is 8, and the sum

The difference between the first and second terms of the geometric progression is 8, and the sum of the second and third terms is 12. Find the first term and denominator of the progression.

Let us denote the first term of the progression by b1 and the denominator of the progression q.

The difference between the first and second members of the progression is 8:

b1 – b1 * q = 8;

The sum of the second and third members of the progression is 12:

b1 * q + b1 * q ^ 2 = 12;

Express q from the first equation and substitute it into the second equation:

q = (b1 – 8) / b1;

b1 * (b1 – 8) / b1 + b1 * ((b1 – 8) / b1) ^ 2 = 12;

b1 – 8 + (b1 ^ 2 – 16b1 + 64) / b1 = 12;

b1 ^ 2 – 8b1 + b1 ^ 2 – 16b1 + 64 – 12b1 = 0;

2b1 ^ 2 – 36b1 + 64 = 0;

b1 ^ 2 – 18b1 + 32 = 0;

By the Vieta converse theorem, b11 = 2 or b12 = 16.

Then the denominator of the progression is:

q1 = (2 – 8) / 2 = – 3;

q2 = (16 – 8) / 16 = 0.5.

Answer: the first term is 2 and the denominator is 3; the first term is 16 and the denominator is 0.5.