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

Key Concepts

Odd Integers

Odd integers are numbers that are not divisible by 2, typically represented in the form of 2n - 1, where n is a positive integer. The sequence of the first n positive odd integers starts from 1 and continues as 1, 3, 5, ..., up to (2n - 1). Understanding this sequence is crucial for analyzing their sum.

Summation of Series

The summation of a series involves adding a sequence of numbers together. In this case, we are summing the first n odd integers. Recognizing the pattern in the series helps in deriving a formula for the sum, which in this case is shown to equal n².

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite number of statements. It involves two steps: proving the base case and then showing that if the statement holds for n, it also holds for n + 1. This method can be applied to prove that the sum of the first n odd integers equals n².