Consider the statement `P(n): n^(2) ge 100`. Here, `P(n) Rightarrow P(n+1)` for all `n`. We can conclude that:

Options

A. (a) `P(n)` is true for all `n`
B. (b) `P(n)` is true for all `n ge 2`
C. (c) `P(n)` is true for all `n ge 3`
D. (d) None of these

Answer

To solve the problem, we need to analyze the statement \( P(n): n^2 \geq 100 \) and determine the validity of the implications given in the options.

### Step 1: Understanding the statement \( P(n) \)
The statement \( P(n) \) asserts that \( n^2 \) is greater than or equal to 100. We need to find the values of \( n \) for which this statement holds true.

### Step 2: Finding the critical value of \( n \)
To find the smallest integer \( n \) such that \( n^2 \geq 100 \), we solve the inequality:
\[ n^2 \geq 100 \]
Taking the square root of both sides gives:
\[ n \geq 10 \quad \text{or} \quad n \leq -10 \]
Since we are typically interested in non-negative integers for this type of problem, we focus on:
\[ n \geq 10 \]

### Step 3: Testing values of \( n \)
Now, let's test some values of \( n \):
- For \( n = 2 \):
\[ P(2): 2^2 = 4 \quad \text{(not } \geq 100\text{, so } P(2) \text{ is false)} \]
- For \( n = 3 \):
\[ P(3): 3^2 = 9 \quad \text{(not } \geq 100\text{, so } P(3) \text{ is false)} \]
- For \( n = 10 \):
\[ P(10): 10^2 = 100 \quad \text{(is } \geq 100\text{, so } P(10) \text{ is true)} \]

### Step 4: Analyzing the implications \( P(n) \Rightarrow P(n+1) \)
The statement \( P(n) \Rightarrow P(n+1) \) means that if \( P(n) \) is true, then \( P(n+1) \) must also be true. Since we have established that \( P(n) \) is only true for \( n \geq 10 \), this implication holds for all \( n \geq 10 \).

### Step 5: Evaluating the options
Now we evaluate the options given in the question:
- **Option A**: \( P(n) \) is true for all \( n \). (False, as shown by testing \( n = 2 \) and \( n = 3 \))
- **Option B**: \( P(n) \) is true for all \( n \geq 2 \). (False, as shown by testing \( n = 2 \) and \( n = 3 \))
- **Option C**: \( P(n) \) is true for all \( n \geq 3 \). (False, as shown by testing \( n = 3 \))
- **Option D**: None of these. (True, since all previous options are false)

### Conclusion
The correct answer is **Option D: None of these**.