If {x} denotes the fractional part of x, then `{(3^(2n))/8},n in N ,` is

Options

A. `(3)/(8) `
B. `(7)/(8)`
C. `(1)/(8)`
D. none of these

Explanation

To solve the problem of finding the fractional part of \(\frac{3^{2n}}{8}\) where \(n\) is a natural number, we can follow these steps:

### Step 1: Rewrite \(3^{2n}\)
We start by rewriting \(3^{2n}\) as:
\[ 3^{2n} = (3^2)^n = 9^n \]

### Step 2: Express \(9^n\) in terms of 8
Next, we can express \(9\) in a form that includes \(8\):
\[ 9 = 8 + 1 \]
Thus,
\[ 9^n = (8 + 1)^n \]

### Step 3: Apply the Binomial Theorem
Using the Binomial Theorem, we can expand \((8 + 1)^n\):
\[ (8 + 1)^n = \sum_{k=0}^{n} \binom{n}{k} 8^k \cdot 1^{n-k} = \sum_{k=0}^{n} \binom{n}{k} 8^k \]

### Step 4: Divide by 8
Now, we need to divide \(9^n\) by \(8\):
\[ \frac{9^n}{8} = \frac{1}{8} \sum_{k=0}^{n} \binom{n}{k} 8^k \]

### Step 5: Separate the integer and fractional parts
The term \(\frac{1}{8} \sum_{k=1}^{n} \binom{n}{k} 8^k\) represents the integer part, since all terms \(k \geq 1\) will yield integers when multiplied by \(8^k\). The only term that does not contribute to the integer part is when \(k=0\):
\[ \frac{1}{8} \cdot \binom{n}{0} \cdot 8^0 = \frac{1}{8} \]

### Step 6: Identify the fractional part
Thus, the fractional part of \(\frac{9^n}{8}\) is:
\[ \text{Fractional part} = \frac{1}{8} \]

### Conclusion
Therefore, the fractional part of \(\frac{3^{2n}}{8}\) is:
\[ \{ \frac{3^{2n}}{8} \} = \frac{1}{8} \]