In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n. 1 + 3 + 5 + ... + (2n - 1) = n^2

Verified step by step guidance

1. **Step 1: Base Case** - Verify the statement for the initial value, usually n = 1. Substitute n = 1 into the left side of the equation: 1 = 1^2. Check if both sides are equal.
2. **Step 2: Inductive Hypothesis** - Assume the statement is true for some positive integer k, i.e., 1 + 3 + 5 + ... + (2k - 1) = k^2.
3. **Step 3: Inductive Step** - Prove the statement for n = k + 1. Start with the left side of the equation for n = k + 1: 1 + 3 + 5 + ... + (2k - 1) + (2(k + 1) - 1).
4. **Step 4: Simplify the Inductive Step** - Use the inductive hypothesis to replace the sum up to (2k - 1) with k^2, then add the next term: k^2 + (2k + 1).
5. **Step 5: Show Equality** - Simplify k^2 + (2k + 1) to (k + 1)^2, thus proving the statement for n = k + 1. This completes the induction.

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite sequence of statements. It consists of two main steps: the base case, where the statement is verified for the initial value (usually n=1), and the inductive step, where one assumes the statement is true for n=k and then proves it for n=k+1. This method is particularly useful for proving formulas involving integers.

Sum of Odd Numbers

The sum of the first n odd numbers can be expressed as 1 + 3 + 5 + ... + (2n - 1). This sequence can be shown to equal n^2, meaning that the sum of the first n odd integers results in a perfect square. Understanding this relationship is crucial for applying mathematical induction to prove the given statement.

Base Case and Inductive Step

In mathematical induction, the base case is the initial step that verifies the statement for the smallest integer, typically n=1. The inductive step involves assuming the statement holds for an arbitrary positive integer n=k and then demonstrating that it must also hold for n=k+1. Both components are essential for completing a valid proof by induction.