We use the formula for the nth term of the geometric progression bn = b1 * qn – 1,
where b1 is the first term of the geometric progression, q is the denominator of the geometric progression.
According to the problem statement, in this geometric progression b4 = 8, b8 = 128.
Applying the formula for the nth term of the geometric progression for n = 4 and n = 8, we obtain the following relations:
b1 * q4 – 1 = 8;
b1 * q8 – 1 = 128.
Dividing the second ratio by the first, we get:
q7 / q3 = 128/8;
q4 = 16;
q4 = 24;
q1 = -2;
q2 = 2.
Substituting the found value of q into the ratio b1 * q3 = 8, we find b1.
For q = -2:
b1 = 8 / q3 = 8 / (-2) 3 = 8 / (-8) = -1.
For q = 2:
b1 = 8 / q3 = 8/23 = 8/8 = 1.
Answer: the first term and denominator of this geometric progression can have two pairs of values: b1 = 1, q = 2 and b1 = -1, q = -2.
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