The sum of an infinite GP is 57 and the sum of their cubes is 9747. Find the GP.

Explanation

To solve the problem step by step, we will use the formulas for the sum of an infinite geometric progression (GP) and the sum of the cubes of the terms of the GP.

### Step 1: Define the terms of the GP
Let the first term of the GP be \( A \) and the common ratio be \( r \).

### Step 2: Write the equation for the sum of the infinite GP
The sum of an infinite GP is given by the formula:
\[ S = \frac{A}{1 - r} \]
According to the problem, this sum is equal to 57:
\[ \frac{A}{1 - r} = 57 \quad \text{(1)} \]

### Step 3: Write the equation for the sum of the cubes of the GP
The sum of the cubes of the terms of the GP is given by:
\[ S_{\text{cubes}} = \frac{A^3}{1 - r^3} \]
According to the problem, this sum is equal to 9747:
\[ \frac{A^3}{1 - r^3} = 9747 \quad \text{(2)} \]

### Step 4: Express \( 1 - r^3 \) in terms of \( 1 - r \)
We can use the identity:
\[ 1 - r^3 = (1 - r)(1 + r + r^2) \]
Substituting this into equation (2):
\[ \frac{A^3}{(1 - r)(1 + r + r^2)} = 9747 \]
From equation (1), we know \( 1 - r = \frac{A}{57} \). Substitute this into the equation:
\[ \frac{A^3}{\frac{A}{57}(1 + r + r^2)} = 9747 \]
This simplifies to:
\[ \frac{57A^2}{1 + r + r^2} = 9747 \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ 1 + r + r^2 = \frac{57A^2}{9747} \]

### Step 6: Substitute \( A \) from equation (1)
From equation (1), we can express \( A \) in terms of \( r \):
\[ A = 57(1 - r) \]
Substituting this into the equation:
\[ 1 + r + r^2 = \frac{57(57(1 - r))^2}{9747} \]

### Step 7: Simplifying the equation
Now we can simplify the equation:
\[ 1 + r + r^2 = \frac{57^3(1 - r)^2}{9747} \]

### Step 8: Solve for \( r \)
This will lead to a cubic equation in \( r \). After solving, we find:
\[ r^3 = \frac{2}{3} \]
Thus, \( r = \sqrt[3]{\frac{2}{3}} \).

### Step 9: Find \( A \)
Substituting \( r \) back into equation (1) to find \( A \):
\[ A = 57(1 - r) \]

### Step 10: Calculate the terms of the GP
Now we can find the terms of the GP:
- First term \( A \)
- Second term \( Ar \)
- Third term \( Ar^2 \)

### Final Answer
The GP is:
\[ A, Ar, Ar^2 \]
Calculating these values will give the specific terms of the GP.