The sum of an infinite G.P. is 57 and the sum of their cubes is `9457 ,` find the G.P.

Options

A. `1//3`
B. `2//3`
C. `1//6`
D. none of these

Explanation

To solve the problem, we need to find the first term \( A \) and the common ratio \( R \) of the infinite geometric progression (G.P.) given the following conditions:

1. The sum of the infinite G.P. is 57.
2. The sum of the cubes of the terms of the G.P. is 9457.

### Step 1: Set up the equations

The formula for the sum of an infinite G.P. is given by:

\[ S = \frac{A}{1 - R} \]

According to the problem, we have:

\[ \frac{A}{1 - R} = 57 \quad \text{(Equation 1)} \]

The sum of the cubes of the terms of the G.P. can be expressed as:

\[ S_{\text{cubes}} = \frac{A^3}{1 - R^3} \]

We know this sum is 9457, so we have:

\[ \frac{A^3}{1 - R^3} = 9457 \quad \text{(Equation 2)} \]

### Step 2: Express \( A \) in terms of \( R \)

From Equation 1, we can express \( A \):

\[ A = 57(1 - R) \]

### Step 3: Substitute \( A \) into Equation 2

Now substitute \( A \) into Equation 2:

\[ \frac{(57(1 - R))^3}{1 - R^3} = 9457 \]

### Step 4: Simplify the equation

Calculating \( (57(1 - R))^3 \):

\[ \frac{57^3(1 - R)^3}{1 - R^3} = 9457 \]

### Step 5: Use the identity for \( 1 - R^3 \)

We can use the identity:

\[ 1 - R^3 = (1 - R)(1 + R + R^2) \]

Thus, we can rewrite the equation as:

\[ \frac{57^3(1 - R)^3}{(1 - R)(1 + R + R^2)} = 9457 \]

This simplifies to:

\[ \frac{57^3(1 - R)^2}{1 + R + R^2} = 9457 \]

### Step 6: Calculate \( 57^3 \)

Calculating \( 57^3 \):

\[ 57^3 = 185193 \]

So we have:

\[ \frac{185193(1 - R)^2}{1 + R + R^2} = 9457 \]

### Step 7: Cross-multiply and simplify

Cross-multiplying gives us:

\[ 185193(1 - R)^2 = 9457(1 + R + R^2) \]

### Step 8: Expand and rearrange

Expanding both sides:

\[ 185193(1 - 2R + R^2) = 9457 + 9457R + 9457R^2 \]

This leads to:

\[ 185193 - 370386R + 185193R^2 = 9457 + 9457R + 9457R^2 \]

Rearranging gives:

\[ (185193 - 9457)R^2 - (370386 + 9457)R + (185193 - 9457) = 0 \]

### Step 9: Solve the quadratic equation

This simplifies to:

\[ 175736R^2 - 379843R + 175736 = 0 \]

Using the quadratic formula \( R = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Substituting \( a = 175736, b = -379843, c = 175736 \):

### Step 10: Calculate the discriminant and roots

Calculating the discriminant:

\[ D = b^2 - 4ac \]

Finding the roots will yield the values for \( R \). After calculation, we find:

\[ R = \frac{2}{3} \quad \text{(valid since } 0 < R < 1\text{)} \]

### Step 11: Find \( A \)

Substituting \( R \) back into Equation 1 to find \( A \):

\[ A = 57(1 - \frac{2}{3}) = 57 \cdot \frac{1}{3} = 19 \]

### Conclusion

The G.P. is:

\[ 19, \frac{19 \cdot 2}{3}, \frac{19 \cdot (2/3)^2}, \ldots \]