For a GP, If `S_(n)=(4^(n)-3^(n))/(3^(n)),` then `t_(2)=` . . .

Options

A. `(1)/(9)`
B. `(2)/(9)`
C. `(7)/(9)`
D. `(4)/(9)`

Explanation

To find the second term \( T_2 \) of the geometric progression (GP) given the sum of the first \( n \) terms \( S_n = \frac{4^n - 3^n}{3^n} \), we can follow these steps:

### Step 1: Understand the relationship between \( S_n \) and the terms of the GP
The sum of the first \( n \) terms \( S_n \) can be expressed as:
\[ S_n = T_1 + T_2 + T_3 + \ldots + T_n \]
Where \( T_n \) is the \( n \)-th term of the GP.

### Step 2: Calculate \( S_1 \) to find \( T_1 \)
Substituting \( n = 1 \) into the formula for \( S_n \):
\[ S_1 = \frac{4^1 - 3^1}{3^1} = \frac{4 - 3}{3} = \frac{1}{3} \]
Since \( S_1 = T_1 \), we have:
\[ T_1 = \frac{1}{3} \]

### Step 3: Calculate \( S_2 \) to find \( T_2 \)
Now, substituting \( n = 2 \) into the formula for \( S_n \):
\[ S_2 = \frac{4^2 - 3^2}{3^2} = \frac{16 - 9}{9} = \frac{7}{9} \]
From the relationship of the sums, we know:
\[ S_2 = T_1 + T_2 \]
Substituting the known value of \( T_1 \):
\[ S_2 = \frac{1}{3} + T_2 \]
Thus, we can set up the equation:
\[ \frac{7}{9} = \frac{1}{3} + T_2 \]

### Step 4: Solve for \( T_2 \)
To solve for \( T_2 \), we need to express \( \frac{1}{3} \) with a common denominator of 9:
\[ \frac{1}{3} = \frac{3}{9} \]
Now substituting this into the equation:
\[ \frac{7}{9} = \frac{3}{9} + T_2 \]
Subtract \( \frac{3}{9} \) from both sides:
\[ T_2 = \frac{7}{9} - \frac{3}{9} = \frac{4}{9} \]

### Final Answer
Thus, the second term \( T_2 \) is:
\[ T_2 = \frac{4}{9} \]