In an exponential progression, all the terms of which are positive, the sum of the first two is 8

In an exponential progression, all the terms of which are positive, the sum of the first two is 8, and the sum of the third and fourth terms is 72. How many members of this progression, starting with the first, must be added to get the total of 242?

We use the ratio:
bn + 1 = bn * q, where q is the denominator:

b2 = b1 * q;
b3 = b2 * q = b1 * q ^ 2;
b4 = b3 * q = b1 * q ^ 3;

b1 + b2 = 8;
b3 + b4 = 72;

b1 + b1 * q = 8;
b1 * q ^ 2 + b1 * q ^ 3 = 72;

b1 * (1 + q) = 8;
b1 * q ^ 2 (1 + q) = 72;

We divide the last equation by the penultimate one:

q ^ 2 = 9;
q = 3;

b1 (1 + 3) = 8;
b1 = 2;

S = b1 (1 – q ^ n) / (1 – q);
242 = 2 (1 – q ^ n) / (1 – 3);
-484 = 2 (1 – 3 ^ n);
-484 = 2 – 2 * 3 ^ n;

486 = 2 * 3 ^ n;
243 = 3 ^ n;
3 ^ 5 = 3 ^ n;
n = 5.

Answer. You need to add 5 members of a geometric progression.

Check: 2 + 6 + 18 + 54 + 162 = 242.