Final answer:
The sum of the first n positive odd numbers equals n^2, which is a result often attributed to Carl Friedrich Gauss. The proof involves pairing terms in an arithmetic series of odd numbers and showing that each pair sums to 2n, leading to a total sum of 2n^2, which when divided by 2, yields n^2.
Step-by-step explanation:
The question refers to the famous proof for the sum of the first n positive odd numbers, which equals n2. This result was famously demonstrated by mathematicians but is often attributed to Carl Friedrich Gauss, who as a child, found the sum of an arithmetic series by a clever method (though his method dealt with even numbers, it has been extended to apply to similar problems). To understand the sum of odd numbers specifically, consider an arithmetic series of odd numbers starting with 1: 1, 3, 5, ..., (2n - 3), (2n - 1). In this series, each term can be written as 2i - 1 where i ranges from 1 to n. To prove the sum equals n2, you can play a rearranging game.
Pair the first and last terms and observe that 1 + (2n - 1) simplifies to 2n. If you continue pairing terms together in this manner, you'll find that each pair sums to 2n. Since you have n pairs, the total sum will be n multiplied by 2n, which gives 2n2. However, because you're adding each number twice, you need to divide the total by 2 to get the actual sum, resulting in n2. Thus, the sum of the first n odd numbers is indeed n2.