Mathematical Induction Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Mathematical Induction.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Mathematical induction proves statements indexed by integers by verifying a base case and an inductive step.

Like dominoes: first one falls, and each one knocks over the next.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Base truth plus step implication yields truth for all integers in range.

Common stuck point: Students prove the step for n+1 without correctly using the assumption for n.

Sense of Study hint: Write the induction hypothesis explicitly before manipulating the n+1 case.

Worked Examples

Example 1

medium
Use mathematical induction to prove: 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2} for all n \ge 1.

Solution

  1. 1
    Base case (n=1): LHS = 1. RHS = \frac{1 \cdot 2}{2} = 1. True.
  2. 2
    Inductive hypothesis: Assume the formula holds for n = k, i.e., 1 + 2 + \cdots + k = \frac{k(k+1)}{2}.
  3. 3
    Inductive step: Show it holds for n = k+1. LHS = \frac{k(k+1)}{2} + (k+1) = \frac{k(k+1) + 2(k+1)}{2} = \frac{(k+1)(k+2)}{2}.
  4. 4
    This matches \frac{(k+1)((k+1)+1)}{2}. By induction, the formula holds for all n \ge 1.

Answer

1 + 2 + \cdots + n = \frac{n(n+1)}{2} \text{ for all } n \ge 1
Mathematical induction has two steps: verify a base case, then show that if the statement holds for k, it holds for k+1. This creates a chain of truth for all natural numbers.

Example 2

hard
Prove by induction: n! > 2^n for all integers n \ge 4.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Prove by induction: 1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6} for all n \ge 1.

Example 2

medium
Prove by induction that 2 + 4 + 6 + \cdots + 2n = n(n+1) for all n \ge 1.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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