Problem: The sum of the first two terms of an infinite geometric series is 15 and each term is equal to the sum of all the terms following it. Find the sum of the series.

Solution:

We are given that the first two terms of the series add up to 15. Let these terms be a and ar, where r is the common ratio.

Step 1: Find the common ratio

Since each term is equal to the sum of all the terms following it, we can write:

a = ar + ar^2 + ar^3 + ...
Dividing both sides by a, we get:
1 = r + r^2 + r^3 + ...

This is a geometric series with first term r and common ratio r. We know that the sum of an infinite geometric series is given by:
S = a / (1 - r)

So, we can find the sum of the series in terms of r:

S = ar / (1 - r)

Step 2: Use the given information to form an equation

We know that a + ar = 15, so we can substitute a = 15 - ar into the expression for S:

S = ar / (1 - r)
S = (15 - ar)r / (1 - r)

Step 3: Solve for r

We can solve for r by manipulating the expression for S:

S = (15 - ar)r / (1 - r)
S(1 - r) = (15 - ar)r
S - Sr = 15r - ar^2
ar^2 + (S - 15)r - S = 0

This is a quadratic equation in r. We can solve for r using the quadratic formula:

r = [-(S - 15) ± sqrt((S - 15)^2 + 4Sa)] / (2a)

Step 4: Find the sum of the series

Once we have found r, we can use the expression for S to find the sum of the series:

S = ar / (1 - r)

Final Answer: The sum of the series is obtained by plugging the value of r into the expression for S:

S = ar / (1 - r)

Note: The final answer will depend on the values of a and r, which can be found using the quadratic formula.