Integral part of `(7 + 4sqrt3)^n` is `(n in N)`

Options

A. an even number
B. an odd number
C. an even or an odd number depending upon the value of n
D. nothing can be said

Answer

To find the integral part of \((7 + 4\sqrt{3})^n\) where \(n\) is a natural number, we can follow these steps:

### Step 1: Identify the Expression
We start with the expression \((7 + 4\sqrt{3})^n\). We can also consider its conjugate, which is \((7 - 4\sqrt{3})^n\).

### Step 2: Analyze the Conjugate
The term \((7 - 4\sqrt{3})\) is a positive number less than 1, since \(4\sqrt{3} \approx 6.928\) and \(7 - 4\sqrt{3} \approx 0.072\). Therefore, as \(n\) increases, \((7 - 4\sqrt{3})^n\) approaches 0.

### Step 3: Use Binomial Theorem
Using the Binomial Theorem, we can expand \((7 + 4\sqrt{3})^n\) as follows:
\[ (7 + 4\sqrt{3})^n = \sum_{k=0}^{n} \binom{n}{k} 7^{n-k} (4\sqrt{3})^k \]

### Step 4: Separate Integral and Non-Integral Parts
The integral part of \((7 + 4\sqrt{3})^n\) can be expressed as:
\[ \text{Integral part} = (7 + 4\sqrt{3})^n + (7 - 4\sqrt{3})^n - (7 - 4\sqrt{3})^n \]
Since \((7 - 4\sqrt{3})^n\) is very small, we can approximate:
\[ \text{Integral part} \approx (7 + 4\sqrt{3})^n \]

### Step 5: Determine the Integral Part
The integral part of \((7 + 4\sqrt{3})^n\) is essentially the integer part of the expression since the second term is negligible. Thus, the integral part is:
\[ \lfloor (7 + 4\sqrt{3})^n \rfloor \]

### Step 6: Conclude the Result
Since \((7 + 4\sqrt{3})^n\) is a sum of an integer and a very small positive number, the integral part will always be an integer.

Thus, the integral part of \((7 + 4\sqrt{3})^n\) is an odd number for all \(n \in \mathbb{N}\).

### Final Answer
The integral part of \((7 + 4\sqrt{3})^n\) is an odd number.

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