Mathematical induction is a proof technique used in mathematics to establish the truth of an infinite sequence of statements, typically indexed by the natural numbers $n$. It involves two main steps: the base case and the inductive step.

Here’s how mathematical induction works:

---

1. Base Case

Prove that the statement holds for the first value, often $n = 1$ (or $n = 0$, depending on the problem). This verifies that the statement is true for the starting point of the sequence.

---

2. Inductive Hypothesis

Assume the statement holds for some arbitrary positive integer $k$. This assumption is called the inductive hypothesis. For example, if the statement is $P(n)$, assume that $P(k)$ is true.

---

3. Inductive Step

Using the inductive hypothesis, prove that the statement is true for $k+1$. In other words, show that: $P(k) \implies P(k+1)$ This demonstrates that if the statement is true for $k$, then it must also be true for $k+1$.

---

Conclusion

If both the base case and the inductive step are valid, then by the principle of mathematical induction, the statement is true for all $n$ starting from the base case.

---

Example: Prove $1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}$

Step 1: Base Case

For $n = 1$: $1 = \frac{1(1+1)}{2} = 1$ The base case holds.

Step 2: Inductive Hypothesis

Assume the formula holds for $n = k$, i.e., $1 + 2 + \dots + k = \frac{k(k+1)}{2}.$

Step 3: Inductive Step

Prove the formula for $n = k+1$: $1 + 2 + \dots + k + (k+1) = \frac{k(k+1)}{2} + (k+1).$ Simplify the right-hand side: $\frac{k(k+1)}{2} + (k+1) = \frac{k(k+1) + 2(k+1)}{2} = \frac{(k+1)(k+2)}{2}.$ Thus, the formula holds for $k+1$.

Conclusion

By the principle of mathematical induction, $1 + 2 + \dots + n = \frac{n(n+1)}{2}$ is true for all $n \geq 1$.

---

Would you like details on a specific problem or additional examples?

5 Related Questions:

1. How does mathematical induction differ from other proof techniques like contradiction?
2. Can mathematical induction be applied to inequalities?
3. What are the limitations of mathematical induction?
4. How do you handle problems where the base case is not $n = 1$?
5. Are there variations of mathematical induction, such as strong induction?

Tip:

Always clearly label each step (base case, inductive hypothesis, inductive step) to avoid confusion and ensure logical flow in your proof.