In a geometric progression, the difference between the sixth and fourth terms is 24

In a geometric progression, the difference between the sixth and fourth terms is 24, and the difference between the third and fifth terms is 12. Find the sum of the first eight terms of this progression.

To solve, we use the formula bn = b1 * q ^ (n – 1), then we get a system of two equations:

b1 * q ^ 5 – b1 * q ^ 3 = 24

b1 * q ^ 4 – b1 * b ^ 2 = 12

Let us express b1 from the 1st equation and substitute it into the 2nd:

24 * q ^ 4 / (q ^ 5 – q ^ 3) – 24 * q ^ 2 / (q ^ 5 – q ^ 3) = 12

2 * q ^ 2 – 2 = q ^ 3 – q

q ^ 3 -2q ^ 2 + q – 2 = 0

q = 2.

Find b1:

b1 * 2 ^ 4 – b1 * 2 ^ 2 = 12

b1 * 12 = 12

b1 = 1.

We find the sum of n terms by the formula:

Sn = b1 * (1 – q ^ n) / (1 – q).

S8 = 1 * (1 – 2 ^ 8) / (1 – 2) = 256 -1 = 255