If the sum of an infinite G.P. is `(7)/(2)` and sum of the squares of its terms is `(147)/(16)` then the sum of the cubes of its terms is

Options

A. `(315)/(19)`
B. `(700)/(39)`
C. `(985)/(13)`
D. `(1029)/(38)`

Answer

To solve the problem step by step, we will use the properties of a geometric progression (G.P.) and the formulas for the sums of the series.

### Step 1: Understand the given information
We are given:
1. The sum of an infinite G.P. \( S = \frac{7}{2} \)
2. The sum of the squares of its terms \( S_2 = \frac{147}{16} \)

### Step 2: Use the formula for the sum of an infinite G.P.
The sum of an infinite G.P. is given by the formula:
\[ S = \frac{A}{1 - r} \]
where \( A \) is the first term and \( r \) is the common ratio. From the given information, we can write:
\[ \frac{A}{1 - r} = \frac{7}{2} \tag{1} \]

### Step 3: Use the formula for the sum of the squares of the terms
The sum of the squares of the terms of an infinite G.P. is given by:
\[ S_2 = \frac{A^2}{1 - r^2} \]
From the problem, we have:
\[ \frac{A^2}{1 - r^2} = \frac{147}{16} \tag{2} \]

### Step 4: Solve equations (1) and (2)
We will first express \( A \) from equation (1):
\[ A = \frac{7}{2}(1 - r) \tag{3} \]

Now, substitute equation (3) into equation (2):
\[ \frac{\left(\frac{7}{2}(1 - r)\right)^2}{1 - r^2} = \frac{147}{16} \]
This simplifies to:
\[ \frac{\frac{49}{4}(1 - r)^2}{1 - r^2} = \frac{147}{16} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 49(1 - r)^2 \cdot 16 = 147 \cdot 4(1 - r^2) \]
This simplifies to:
\[ 784(1 - r)^2 = 588(1 - r^2) \]

### Step 6: Expand both sides
Expanding both sides:
\[ 784(1 - 2r + r^2) = 588(1 - r^2) \]
This leads to:
\[ 784 - 1568r + 784r^2 = 588 - 588r^2 \]

### Step 7: Combine like terms
Rearranging gives:
\[ (784 + 588)r^2 - 1568r + (784 - 588) = 0 \]
which simplifies to:
\[ 1372r^2 - 1568r + 196 = 0 \]

### Step 8: Solve the quadratic equation
Using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1372 \), \( b = -1568 \), and \( c = 196 \):
\[ r = \frac{1568 \pm \sqrt{(-1568)^2 - 4 \cdot 1372 \cdot 196}}{2 \cdot 1372} \]

### Step 9: Calculate the value of \( r \)
Calculating the discriminant and solving for \( r \) will give us the value of \( r \).

### Step 10: Find \( A \) using \( r \)
Once we have \( r \), we can substitute it back into equation (3) to find \( A \).

### Step 11: Calculate the sum of cubes
The sum of the cubes of the terms is given by:
\[ S_3 = \frac{A^3}{1 - r^3} \]
Substituting the values of \( A \) and \( r \) will give us the final answer.

### Final Answer
After performing the calculations, we find that the sum of the cubes of the terms is:
\[ \frac{1029}{38} \]