Does the Principle of Mathematical Induction' apply to :
`P(n) : n^(3) gt 100` ?

Answer

To determine whether the Principle of Mathematical Induction applies to the statement \( P(n): n^3 > 100 \), we will follow the steps of mathematical induction and analyze the base case and the inductive step.

### Step 1: Define the statement
We want to check if the statement \( P(n): n^3 > 100 \) holds for all natural numbers \( n \).

### Step 2: Base Case
We first check the base case, which is typically \( n = 1 \).

\[ P(1): 1^3 = 1 \quad \text{(which is not greater than 100)} \]

Since \( P(1) \) is false, the base case fails.

### Step 3: Inductive Step
Even though the base case has already failed, let's assume \( P(k) \) is true for some \( k \) (i.e., \( k^3 > 100 \)). We would then need to show that \( P(k + 1) \) is also true (i.e., \( (k + 1)^3 > 100 \)).

Calculating \( (k + 1)^3 \):

\[ (k + 1)^3 = k^3 + 3k^2 + 3k + 1 \]

We need to check if:

\[ k^3 + 3k^2 + 3k + 1 > 100 \]

Given that \( k^3 > 100 \), we can analyze this for various values of \( k \):

- For \( k = 4 \):
\[ 4^3 = 64 \quad \text{(not greater than 100)} \]

- For \( k = 5 \):
\[ 5^3 = 125 \quad \text{(greater than 100)} \]
\[ (5 + 1)^3 = 6^3 = 216 \quad \text{(greater than 100)} \]

From this analysis, we see that \( P(k) \) is true for \( k = 5 \) and greater, but not for \( k = 1, 2, 3, 4 \).

### Conclusion
Since the base case \( P(1) \) is false, the statement \( P(n): n^3 > 100 \) does not hold for all natural numbers \( n \). Therefore, the Principle of Mathematical Induction does not apply to this statement.