In an infinite geometric progression, the sum of first two terms is 6 and every terms is four times the sum of all the terms that it . Find :
(i) the geometric progression and (ii) its sum to infinity .

Answer

To solve the problem step by step, we will follow the information given in the question and use the properties of geometric progressions (GP).

### Given:
1. The sum of the first two terms is 6.
2. Every term is four times the sum of all the terms that come after it.

### Let:
- The first term of the GP be \( a \).
- The common ratio be \( r \).

### Step 1: Write the equation for the sum of the first two terms.
The first two terms of the GP are \( a \) and \( ar \).
According to the problem, their sum is:
\[ a + ar = 6 \]
Factoring out \( a \):
\[ a(1 + r) = 6 \quad \text{(Equation 1)} \]

### Step 2: Write the equation for the relationship between the terms and the sum of the remaining terms.
The sum of the infinite GP can be expressed as:
\[ S = \frac{a}{1 - r} \quad \text{(for } |r| < 1\text{)} \]
The sum of all terms after the first term is:
\[ S' = S - a = \frac{a}{1 - r} - a = \frac{a - a(1 - r)}{1 - r} = \frac{ar}{1 - r} \]
According to the problem, each term \( ar^n \) is four times the sum of all subsequent terms:
\[ ar = 4 \cdot S' = 4 \cdot \frac{ar}{1 - r} \]
This simplifies to:
\[ ar = \frac{4ar}{1 - r} \]
Assuming \( ar \neq 0 \), we can divide both sides by \( ar \):
\[ 1 = \frac{4}{1 - r} \]
Cross-multiplying gives:
\[ 1 - r = 4 \implies r = -3 \quad \text{(Equation 2)} \]

### Step 3: Substitute \( r \) back into Equation 1 to find \( a \).
Substituting \( r = -3 \) into Equation 1:
\[ a(1 - 3) = 6 \implies a(-2) = 6 \implies a = -3 \]

### Step 4: Write the geometric progression.
Now we have:
- \( a = -3 \)
- \( r = -3 \)

The GP can be written as:
\[ -3, -3(-3), -3(-3)^2, -3(-3)^3, \ldots \]
This simplifies to:
\[ -3, 9, -27, 81, \ldots \]

### Step 5: Calculate the sum to infinity.
Using the formula for the sum to infinity:
\[ S = \frac{a}{1 - r} = \frac{-3}{1 - (-3)} = \frac{-3}{1 + 3} = \frac{-3}{4} = -0.75 \]

### Final Answers:
(i) The geometric progression is:
\[ -3, 9, -27, 81, \ldots \]
(ii) The sum to infinity is:
\[ -0.75 \]