The sum of an infinite decreasing geometric progression is 30, and the sum of the fourth powers

The sum of an infinite decreasing geometric progression is 30, and the sum of the fourth powers of its members is 54,000. What is the first term of the progression?

First, we write down the sum of all members of an infinite geometric progression:

b / (1 – q) = 30.

And now four degrees:

b ^ 4 / (1 – q ^ 4) = 54000.

Now, from this whole expression, we express b:

b = 30 – 30q.

Substitute this expression for b:

(30 ^ 4 (1 – q) ^ 4) / 1 – q ^ 4 = 54000.

(1 – q) ^ 4/1 – q ^ 4 = 1/15.

15 (1 – q) ^ 4 = 1 – q ^ 4.

Now we need to expand the brackets and transform the expression:

2 (2q – 1) (q – 1) (4q ^ 2 – 9q + 7) = 0.

q = 0.5; 1.

Now let’s find the first term of the progression:

b = 30 – 30q.

Substitute q1:

b = 30 – 30 * 0.5 = 30 – 15 = 15.