Find the sum of the first eight terms of the geometric progression bn with positive terms knowing

Find the sum of the first eight terms of the geometric progression bn with positive terms knowing that b2 = 1.2 and b4 = 4.8.

According to the condition of the problem, a geometric progression bn is given, in which the second term is b2 = 1.2, and the fourth term is b4 = 4.8.

Using 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, we can write the following ratios:

b1 * q2 – 1 = 1.2;

b1 * q4 – 1 = 4.8.

We solve the resulting system of equations.

Dividing the second equation by the first, we get:

b1 * q3 / (b1 * q) = 4.8 /1.2;

q² = 4;

q² = 2².

According to the condition of the problem, this geometric progression has positive terms, therefore, q = 2.

Substituting the found value of q into the ratio b1 * q = 1.2, we obtain;

b1 * 2 = 1.2;

b1 = 1.2 / 2;

b1 = 0.6.

Using the formula for the sum of the first n terms of the geometric progression Sn = b1 * (1 – qⁿ) / (1 – q) for n = 8, we find the sum of the first eight members of this geometric progression:

S8 = 0.6 * (1 – 28) / (1 – 2) = 0.6 * (1 – 256) / (-1) = 0.6 * 255 = 153.

Answer: the sum of the first eight members of this geometric progression is 153.