The value of the natural numbers n such that `2 ^(n) gt 2n +1 ` is valid for

Options

A. `n ge 3`
B. `n lt 3`
C. `n gt 1`
D. any n

Answer

To solve the inequality \(2^n > 2n + 1\) for natural numbers \(n\), we will evaluate the inequality for different values of \(n\) and then use mathematical induction to establish the validity of the inequality.

### Step-by-Step Solution:

1. **Evaluate for \(n = 1\)**:
\[ 2^1 > 2 \cdot 1 + 1 \implies 2 > 3 \]
This is false.

**Hint**: Check if the left side is greater than the right side for small values of \(n\).

2. **Evaluate for \(n = 2\)**:
\[ 2^2 > 2 \cdot 2 + 1 \implies 4 > 5 \]
This is also false.

**Hint**: Continue testing small natural numbers to find where the inequality holds.

3. **Evaluate for \(n = 3\)**:
\[ 2^3 > 2 \cdot 3 + 1 \implies 8 > 7 \]
This is true.

**Hint**: Look for the first value of \(n\) where the inequality holds.

4. **Evaluate for \(n = 4\)**:
\[ 2^4 > 2 \cdot 4 + 1 \implies 16 > 9 \]
This is true.

**Hint**: Check consecutive values to see if the pattern continues.

5. **Evaluate for \(n = 5\)**:
\[ 2^5 > 2 \cdot 5 + 1 \implies 32 > 11 \]
This is true.

**Hint**: If the inequality holds for several consecutive values, it may hold for larger values as well.

6. **Inductive Step**:
We will prove that if \(2^k > 2k + 1\) holds for some \(k \geq 3\), then it also holds for \(k + 1\).

**Base Case**: We have already established that \(2^3 > 2 \cdot 3 + 1\) is true.

**Inductive Hypothesis**: Assume \(2^k > 2k + 1\) is true for some \(k \geq 3\).

**Inductive Step**:
We need to show that \(2^{k+1} > 2(k+1) + 1\).
\[ 2^{k+1} = 2 \cdot 2^k \]
By the inductive hypothesis:
\[ 2 \cdot 2^k > 2(2k + 1) = 4k + 2 \]
Now, we need to show:
\[ 4k + 2 > 2k + 2 + 1 \implies 4k + 2 > 2k + 3 \]
Simplifying this gives:
\[ 2k > 1 \]
This is true for all \(k \geq 1\).

Therefore, by induction, \(2^n > 2n + 1\) holds for all \(n \geq 3\).

### Conclusion:
The values of natural numbers \(n\) such that \(2^n > 2n + 1\) is valid for \(n \geq 3\).

**Final Answer**: \(n \geq 3\)

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