If the sum of `n` terms of a G.P. is `3(3^(n+1))/(4^(2n))` , then find the common ratio.

Explanation

To find the common ratio of the given geometric progression (G.P.) where the sum of the first \( n \) terms is given by:

\[ S_n = \frac{3(3^{n+1})}{4^{2n}} \]

we will follow these steps:

### Step 1: Find \( S_1 \)
The first term \( T_1 \) of the G.P. is equal to the sum of the first term:

\[ S_1 = \frac{3(3^{1+1})}{4^{2 \cdot 1}} = \frac{3(3^2)}{4^2} = \frac{3 \cdot 9}{16} = \frac{27}{16} \]

### Step 2: Find \( S_2 \)
Next, we calculate the sum of the first two terms \( S_2 \):

\[ S_2 = \frac{3(3^{2+1})}{4^{2 \cdot 2}} = \frac{3(3^3)}{4^4} = \frac{3 \cdot 27}{256} = \frac{81}{256} \]

### Step 3: Find \( T_2 \)
Using the relationship \( S_2 = T_1 + T_2 \), we can express \( T_2 \):

\[ T_2 = S_2 - S_1 = \frac{81}{256} - \frac{27}{16} \]

To perform this subtraction, we need a common denominator. The common denominator of 256 and 16 is 256:

\[ \frac{27}{16} = \frac{27 \cdot 16}{16 \cdot 16} = \frac{432}{256} \]

Now substituting back:

\[ T_2 = \frac{81}{256} - \frac{432}{256} = \frac{81 - 432}{256} = \frac{-351}{256} \]

### Step 4: Find \( R \) (Common Ratio)
The common ratio \( R \) can be calculated using the formula:

\[ R = \frac{T_2}{T_1} \]

We already have \( T_1 = \frac{27}{16} \) and \( T_2 = \frac{-351}{256} \):

\[ R = \frac{T_2}{T_1} = \frac{-351/256}{27/16} = \frac{-351}{256} \cdot \frac{16}{27} \]

### Step 5: Simplify \( R \)
Now we simplify this expression:

\[ R = \frac{-351 \cdot 16}{256 \cdot 27} \]

Calculating the numerator and denominator:

\[ R = \frac{-5616}{6912} \]

Now simplifying \( -5616 \div 6912 \):

\[ R = -\frac{3}{4} \]

### Conclusion
Thus, the common ratio \( R \) of the geometric progression is:

\[ \boxed{-\frac{3}{4}} \]