Show that the sum of the first n positive odd integers,1 +3 +5 + ··· + (2n − 1), ... is n².

Verified step by step guidance

1. Recognize the sequence given: the first n positive odd integers are 1, 3, 5, ..., (2n - 1). Each term can be expressed as the general term \(a_k = 2k - 1\) where \(k\) ranges from 1 to \(n\).
2. Write the sum of the first n odd integers as a summation: \(S = \sum_{k=1}^n (2k - 1)\).
3. Use the properties of summation to separate the sum into two parts: \(S = \sum_{k=1}^n 2k - \sum_{k=1}^n 1\).
4. Evaluate each summation separately: \(\sum_{k=1}^n 2k = 2 \sum_{k=1}^n k\) and \(\sum_{k=1}^n 1 = n\). Recall that \(\sum_{k=1}^n k = \frac{n(n+1)}{2}\).
5. Substitute the known formula into the expression and simplify: \(S = 2 \times \frac{n(n+1)}{2} - n = n(n+1) - n = n^2 + n - n = n^2\). This shows that the sum of the first n odd integers is \(n^2\).

Key Concepts

Arithmetic Series

An arithmetic series is the sum of terms in an arithmetic sequence, where each term increases by a constant difference. In this problem, the sequence of odd integers (1, 3, 5, ...) has a common difference of 2, and understanding this helps in summing the terms.

Mathematical Induction

Mathematical induction is a proof technique used to verify statements for all positive integers. It involves proving a base case and then showing that if the statement holds for an integer k, it also holds for k+1. This method is often used to prove formulas like the sum of odd integers equals n².

Sum of Odd Integers Formula

The sum of the first n positive odd integers is given by n². This formula can be derived or proven using induction or by recognizing that adding consecutive odd numbers forms perfect squares, which is a key insight in this problem.