Answer:
Explanation:
To use mathematical induction to prove a statement, we follow three steps: the inductive basis, the inductive hypothesis, and the inductive step.
1. Inductive Basis: We start by proving that the statement is true for the first value in the sequence or set. Let's assume the statement is true for n = 1.
2. Inductive Hypothesis: We assume that the statement is true for some arbitrary value k, which means we assume the statement holds for all values up to k.
3. Inductive Step: Using the assumption from the previous step, we prove that the statement is true for the next value, k + 1.
Let's take an example to illustrate these steps:
Statement: For all positive integers n, the sum of the first n odd numbers is n^2.
1. Inductive Basis: We start by checking the statement for the smallest value, n = 1. The sum of the first odd number, which is 1, is indeed 1^2 = 1. So the statement holds for n = 1.
2. Inductive Hypothesis: Assume the statement is true for some arbitrary value k. This means the sum of the first k odd numbers is k^2.
3. Inductive Step: We need to prove the statement for the next value, k + 1. The sum of the first (k + 1) odd numbers can be expressed as the sum of the first k odd numbers plus the next odd number, which is (k + 1)^2 - k^2 + (2k + 1). Simplifying this expression, we get (k + 1)^2 + 2k + 1 - k^2 = (k + 1)^2 + k + 1.
Now, we can rewrite this as (k + 1)(k + 1) + (k + 1). Factoring out the common factor of (k + 1), we have (k + 1)(k + 2). So, the sum of the first (k + 1) odd numbers is indeed (k + 1)(k + 2), which is equal to (k + 1)^2 + (k + 1).
Therefore, by mathematical induction, we have proven that the statement "the sum of the first n odd numbers is n^2" holds for all positive integers.