The sum of first three terms of a G.P. is `(1)/(8)` of the sum of the next three terms. Find the common ratio of G.P.

Explanation

To solve the problem, we need to find the common ratio of a geometric progression (G.P.) given that the sum of the first three terms is \(\frac{1}{8}\) of the sum of the next three terms.

Let's denote the first term of the G.P. as \(A\) and the common ratio as \(R\). The first three terms of the G.P. can be expressed as:
1. First term: \(A\)
2. Second term: \(AR\)
3. Third term: \(AR^2\)

The next three terms will be:
1. Fourth term: \(AR^3\)
2. Fifth term: \(AR^4\)
3. Sixth term: \(AR^5\)

### Step 1: Write the sum of the first three terms
The sum of the first three terms is:
\[ S_1 = A + AR + AR^2 = A(1 + R + R^2) \]

### Step 2: Write the sum of the next three terms
The sum of the next three terms is:
\[ S_2 = AR^3 + AR^4 + AR^5 = AR^3(1 + R + R^2) \]

### Step 3: Set up the equation based on the problem statement
According to the problem, the sum of the first three terms is \(\frac{1}{8}\) of the sum of the next three terms:
\[ S_1 = \frac{1}{8} S_2 \]
Substituting the expressions we found for \(S_1\) and \(S_2\):
\[ A(1 + R + R^2) = \frac{1}{8} \cdot AR^3(1 + R + R^2) \]

### Step 4: Simplify the equation
Assuming \(1 + R + R^2 \neq 0\), we can divide both sides by \(A(1 + R + R^2)\):
\[ 1 = \frac{1}{8} R^3 \]

### Step 5: Solve for \(R^3\)
Multiplying both sides by 8 gives:
\[ R^3 = 8 \]

### Step 6: Find \(R\)
Taking the cube root of both sides, we find:
\[ R = \sqrt[3]{8} = 2 \]

### Conclusion
The common ratio \(R\) of the geometric progression is \(2\).

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