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The problem you have shared is about proving an inequality through mathematical induction. Here's the extracted problem:

We are asked to prove by **recurrence** (i.e., induction) that:

\[
\forall n \in \mathbb{N}, \quad \left( \frac{a+b}{2} \right)^n \leq \frac{a^n + b^n}{2}
\]

### Steps for Mathematical Induction:

1. **Base Case**: Prove the inequality for \( n = 1 \).
   
   For \( n = 1 \):
   \[
   \left( \frac{a+b}{2} \right)^1 = \frac{a+b}{2}
   \]
   And:
   \[
   \frac{a^1 + b^1}{2} = \frac{a+b}{2}
   \]
   Both sides are equal, so the base case holds.

2. **Inductive Hypothesis**: Assume that the inequality holds for some \( n = k \), i.e., assume:
   \[
   \left( \frac{a+b}{2} \right)^k \leq \frac{a^k + b^k}{2}
   \]

3. **Inductive Step**: Prove that the inequality also holds for \( n = k+1 \). That is, prove:
   \[
   \left( \frac{a+b}{2} \right)^{k+1} \leq \frac{a^{k+1} + b^{k+1}}{2}
   \]
   
   Using the inductive hypothesis and algebraic manipulations (likely involving properties of binomials or inequalities like Jensen's inequality or convexity), we aim to show this step is valid.

### Would you like me to go into further detail regarding the inductive step, or do you have any other questions?

Here are 5 related questions you might consider:
1. How does mathematical induction work for inequalities?
2. Can this inequality be connected to the Arithmetic-Geometric Mean (AM-GM) inequality?
3. How do we handle inequalities with powers when applying induction?
4. What other types of problems are best solved using induction?
5. Can Jensen's inequality be applied to solve this problem?

**Tip**: When working with inequalities, always ensure that your manipulations are valid for the domain (in this case, real numbers or natural numbers).

Prove by induction that: ∀n ∈ ℕ, ( (a + b) / 2 )^n ≤ (a^n + b^n) / 2

This problem demonstrates how to prove the inequality ( (a + b) / 2 )^n ≤ (a^n + b^n) / 2 using mathematical induction. Suitable for undergraduate students, it covers key concepts of inequalities, exponentiation, and the inductive step process.

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