Which of the following is/are correct?

Options

A. `(101^(50)-99^(50))lt100^(50)`
B. `(101)^(50)-100^(50)gt99^(50)`
C. `(1000)^(1000)lt(1001)^(999)`
D. `(1001)^(999)lt(1000)^(1000)`

Answer

To solve the problem of determining which of the options are correct, we will analyze each option step by step. Let's denote the options as follows:

- Option A: \( 101^{50} - 99^{50} < 100^{50} \)
- Option B: \( 101^{50} - 99^{50} > 100^{50} \)
- Option C: \( 1001^{999} < 1000^{1000} \)
- Option D: \( 1001^{999} > 1000^{1000} \)

### Step 1: Analyze Option A
We want to check if \( 101^{50} - 99^{50} < 100^{50} \).

1. Rewrite the expression:
\[ 101^{50} - 99^{50} = (100 + 1)^{50} - (100 - 1)^{50} \]

2. Use the Binomial Theorem to expand both terms:
\[ (100 + 1)^{50} = \sum_{k=0}^{50} \binom{50}{k} 100^{50-k} \cdot 1^k \]
\[ (100 - 1)^{50} = \sum_{k=0}^{50} \binom{50}{k} 100^{50-k} \cdot (-1)^k \]

3. Subtract the two expansions:
\[ 101^{50} - 99^{50} = \sum_{k=0}^{50} \binom{50}{k} 100^{50-k} (1 - (-1)^k) \]
This results in:
- For even \( k \): \( 1 - 1 = 0 \)
- For odd \( k \): \( 1 + 1 = 2 \)

4. Thus, we only consider the odd terms:
\[ = 2 \sum_{k \text{ odd}} \binom{50}{k} 100^{50-k} \]

5. Since all terms in the sum are positive, we conclude that \( 101^{50} - 99^{50} > 0 \).

6. Comparing with \( 100^{50} \):
\[ 101^{50} - 99^{50} \text{ is not less than } 100^{50} \text{, hence Option A is false.} \]

### Step 2: Analyze Option B
We already established that:
\[ 101^{50} - 99^{50} > 0 \]
Now we need to check if it is greater than \( 100^{50} \).

1. Since \( 101^{50} - 99^{50} \) is greater than \( 100^{50} \) as shown in the previous analysis, we conclude:
\[ \text{Option B is correct.} \]

### Step 3: Analyze Option C
We need to check if \( 1001^{999} < 1000^{1000} \).

1. Rewrite the expression:
\[ 1001^{999} = \left(1000 + 1\right)^{999} \]

2. Use the Binomial Theorem:
\[ (1000 + 1)^{999} = \sum_{k=0}^{999} \binom{999}{k} 1000^{999-k} \cdot 1^k \]

3. Compare with \( 1000^{1000} = 1000^{999} \cdot 1000 \).

4. Since the first term of the expansion (when \( k=0 \)) is \( 1000^{999} \) and the subsequent terms are positive, we can see that:
\[ 1001^{999} < 1000^{1000} \text{, hence Option C is correct.} \]

### Step 4: Analyze Option D
We need to check if \( 1001^{999} > 1000^{1000} \).

1. From the previous analysis, we have already established that \( 1001^{999} < 1000^{1000} \).

2. Therefore, Option D is false.

### Conclusion
- **Option A**: False
- **Option B**: Correct
- **Option C**: Correct
- **Option D**: False

### Final Answer
The correct options are **B and C**.