Mathematical induction is a proof technique used to establish that a statement is true for all positive integers. It is particularly useful for proving formulas involving sequences, sums, and inequalities.
Steps of Mathematical Induction
Mathematical induction follows two main steps:
Base Case: Prove that the statement is true for the first integer, usually or .
Inductive Step: Assume the statement is true for some arbitrary positive integer (this is called the inductive hypothesis), and then show that it must also be true for .
If both steps are valid, the principle of mathematical induction guarantees that the statement is true for all positive integers.
Pattern Recognition
Pattern recognition is a useful technique for finding a general formula for the nth term of a sequence. By examining a sequence’s first few terms, we can look for trends and establish a rule.
Example: Finding a Formula for the th Term of a Sequence
Consider the sequence:
Observing the pattern:
The first differences are (increasing by )
The second differences are constant , indicating a quadratic formula.
Assuming a formula of the form:
Substituting known terms, solving for , we find:
This process can be generalized to find formulas for many sequences.
Sums of Powers of Integers
Example 1: Sum of the First Positive Integers
Prove that for all :
Step 1: Base Case
For :
which is true.
Step 2: Inductive Hypothesis
Assume the formula holds for , so:
Step 3: Inductive Step
We must show that the formula holds for :
Using the inductive hypothesis:
Factor out :
This matches the formula for , completing the proof.
Example 2: Sum of a Geometric Series
Prove that for all :
Step 1: Base Case
For :
which is true.
Step 2: Inductive Hypothesis
Assume the formula holds for :
Step 3: Inductive Step
For :
Using the inductive hypothesis:
Rewrite the right-hand side:
Thus, the formula is valid for , completing the proof.
Example 3: Sum of the First Squares
Prove that:
Base Case: For :
True.
Inductive Hypothesis: Assume true for :
Inductive Step: Show it holds for :
Simplifying, we obtain:
This matches the formula, completing the proof.
Sums of Powers of Integers
Applications of Mathematical Induction
Mathematical induction is widely used in:
Number Theory – Proving divisibility rules and properties of prime numbers.
Algebra – Validating formulas for sums and products.
Combinatorics – Establishing counting principles and combinatorial identities.
Computer Science – Proving the correctness of recursive algorithms.