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4. [Induction, 5+13 points] Use mathematical induction to prove the following statements. Make sure to show clearly all three steps of induction (inductive basis, inductive hypothesis and inductive st

User Bassam
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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.

User Mirko
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