We will prove by induction that \( 2^n + 3^n < 6^n \) for all \( n \geq 1 \).

### Step 1: **Base Case**
For \( n = 1 \):
\[
2^1 + 3^1 = 2 + 3 = 5
\]
\[
6^1 = 6
\]
Clearly, \( 5 < 6 \), so the inequality holds for \( n = 1 \).

### Step 2: **Induction Hypothesis**
Assume that the inequality \( 2^n + 3^n < 6^n \) holds for some \( n = k \). That is:
\[
2^k + 3^k < 6^k
\]
We will now prove that the inequality also holds for \( n = k + 1 \).

### Step 3: **Inductive Step**
We need to show that:
\[
2^{k+1} + 3^{k+1} < 6^{k+1}
\]

From the properties of exponents, we know:
\[
6^{k+1} = 6 \cdot 6^k
\]
Now, express \( 2^{k+1} \) and \( 3^{k+1} \) using the same properties:
\[
2^{k+1} = 2 \cdot 2^k \quad \text{and} \quad 3^{k+1} = 3 \cdot 3^k
\]
Thus, the inequality we need to prove becomes:
\[
2 \cdot 2^k + 3 \cdot 3^k < 6 \cdot 6^k
\]

Using the induction hypothesis \( 2^k + 3^k < 6^k \), let's try bounding the left-hand side.

We know that:
\[
2 \cdot 2^k + 3 \cdot 3^k < 2 \cdot 6^k + 3 \cdot 6^k = (2 + 3) \cdot 6^k = 5 \cdot 6^k
\]
Thus, we now need to show:
\[
5 \cdot 6^k < 6 \cdot 6^k
\]
Clearly, this is true because \( 5 < 6 \). Hence:
\[
2^{k+1} + 3^{k+1} < 6^{k+1}
\]

### Step 4: **Conclusion**
By mathematical induction, we have shown that \( 2^n + 3^n < 6^n \) for all \( n \geq 1 \).

Would you like more details on any part of this proof, or do you have any questions?

Here are five related questions:
1. Can you explain how induction works in more detail?
2. What is the significance of the base case in induction?
3. Can this inequality be extended for \( n < 1 \)?
4. How would the proof change if the inequality were \( 2^n + 3^n \leq 6^n \)?
5. Can you use strong induction to prove this inequality?

**Tip:** In proofs by induction, always clearly separate the base case, induction hypothesis, and inductive step to avoid confusion.

Learn how to use mathematical induction to prove the inequality 2^n + 3^n < 6^n for all n ≥ 1. This problem involves concepts of inequalities, exponential functions, and the process of induction, making it ideal for students in Grades 11-12.

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