Understanding the Problem:
We are given that the first term of an infinite geometric progression (G.P.) is 4. Additionally, we are given that each term is equal to 3 times the sum of all the following terms. We need to find the common ratio of this G.P.

Solution:

1. Finding the Relationship:
Let's assume the common ratio of the G.P. is 'r'. The first term is given as 4, so the second term will be '4r', the third term will be '4r^2', and so on.

Now, according to the given condition, each term is equal to 3 times the sum of all the following terms. Let's denote the sum of all the following terms as 'S'.

So, the second term '4r' is equal to 3 times 'S'. We can write this as:
4r = 3S

Similarly, the third term '4r^2' is equal to 3 times 'S'. We can write this as:
4r^2 = 3S

2. Expressing Sum of Following Terms:
To find the sum 'S', we need to consider the sum of the terms starting from the third term. Let's denote this sum as 'S1'.

The sum 'S1' can be expressed as the sum of an infinite geometric series with the first term '4r^2' and the common ratio 'r'. The formula for the sum of an infinite geometric series is given by:

S1 = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.

Substituting the values, we have:
S1 = (4r^2) / (1 - r)

3. Expressing Total Sum:
Now, the total sum 'S' can be expressed as the sum of the second term '4r' and the sum 'S1'. We can write this as:
S = 4r + S1

Substituting the value of 'S1', we have:
S = 4r + (4r^2) / (1 - r)

4. Solving for the Common Ratio:
We know that each term is equal to 3 times the sum of all the following terms. So, we can write:
4r = 3S

Substituting the value of 'S' from the above equation, we have:
4r = 3(4r + (4r^2) / (1 - r))

Simplifying this equation, we get:
4r = 12r + 12r^2 / (1 - r)

Multiplying through by (1 - r) to eliminate the denominator, we have:
4r(1 - r) = 12r(1 - r) + 12r^2

Expanding and rearranging the equation, we get:
4r - 4r^2 = 12r - 12r^2 + 12r^2

Simplifying further, we have:
4r - 4r^2 = 12r

Rearranging the equation, we get:
4r^2 - 16r + 12r = 0

Simplifying, we