The positive integer which is just greater than `(1+0.0001)^(1000)` is equal to

Options

A. 3
B. 4
C. 5
D. 2

Answer

To find the positive integer that is just greater than \((1 + 0.0001)^{1000}\), we can use the Binomial Theorem to approximate the expression.

### Step-by-Step Solution:

1. **Rewrite the Expression**:
\[ (1 + 0.0001)^{1000} \]
can be rewritten as:
\[ (1 + 10^{-4})^{1000} \]

2. **Apply the Binomial Theorem**:
The Binomial Theorem states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
Here, let \(a = 1\) and \(b = 10^{-4}\), and \(n = 1000\). Thus, we have:
\[ (1 + 10^{-4})^{1000} = \sum_{k=0}^{1000} \binom{1000}{k} (10^{-4})^k \]

3. **Calculate the First Few Terms**:
We will calculate the first few terms of the expansion:
- For \(k = 0\):
\[ \binom{1000}{0} (10^{-4})^0 = 1 \]
- For \(k = 1\):
\[ \binom{1000}{1} (10^{-4})^1 = 1000 \cdot 10^{-4} = 0.1 \]
- For \(k = 2\):
\[ \binom{1000}{2} (10^{-4})^2 = \frac{1000 \cdot 999}{2} \cdot 10^{-8} = 499500 \cdot 10^{-8} = 0.004995 \]

4. **Sum the Significant Terms**:
Now, we sum the significant terms:
\[ 1 + 0.1 + 0.004995 \approx 1.104995 \]

5. **Neglect Higher Order Terms**:
The higher order terms (for \(k \geq 3\)) will be very small compared to the first few terms, so we can neglect them for our approximation.

6. **Determine the Positive Integer**:
The value we have approximated is:
\[ (1 + 0.0001)^{1000} \approx 1.104995 \]
The positive integer just greater than \(1.104995\) is:
\[ 2 \]

### Final Answer:
The positive integer which is just greater than \((1 + 0.0001)^{1000}\) is \(2\).