A student was asked to prove a statement P(n) by using the principle of mathematical induction. He proved that `P(n) Rightarrow P(n+1)` for all `n in N` and also that P(4) is true:
On the basis of the above he can conclude that P(n) is true.

Options

A. For all `n in N`
B. For all `n gt 4`
C. For all `n ge 4`
D. For all `n lt 4`.

Answer

To solve the problem using the principle of mathematical induction, we need to follow these steps:

### Step 1: Understand the Principle of Mathematical Induction
The principle of mathematical induction consists of two main steps:
1. **Base Case**: Prove that the statement is true for a specific initial value (usually n=1, n=0, or any other starting point).
2. **Inductive Step**: Prove that if the statement is true for an arbitrary natural number n, then it is also true for n+1.

### Step 2: Identify the Given Information
From the problem, we know:
- The student has shown that \( P(n) \Rightarrow P(n+1) \) for all \( n \in \mathbb{N} \).
- The base case \( P(4) \) is true.

### Step 3: Apply the Inductive Step
Since \( P(4) \) is true, we can use the inductive step:
- From \( P(4) \), we can conclude \( P(5) \) is true because \( P(4) \Rightarrow P(5) \).
- Next, since \( P(5) \) is true, we can conclude \( P(6) \) is true because \( P(5) \Rightarrow P(6) \).
- Continuing this process, we can conclude that \( P(7) \), \( P(8) \), and so on, are also true.

### Step 4: Generalize the Result
From the above steps, we can see that:
- The statement \( P(n) \) is true for all \( n \geq 4 \).
- Therefore, we can conclude that \( P(n) \) is true for all \( n \) such that \( n \) is greater than or equal to 4.

### Conclusion
Thus, the correct conclusion based on the given information is that \( P(n) \) is true for all \( n \geq 4 \).

### Final Answer
The correct option is: **Option 3: for all \( n \geq 4 \)**.

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