Here are the essential concepts you must grasp in order to answer the question correctly.
Arithmetic Series
An arithmetic series is the sum of the terms of an arithmetic sequence, where each term increases by a constant difference. In this case, the series 1 + 3 + 5 + ... + (2n - 1) consists of the first n odd numbers, which can be expressed as an arithmetic series with a common difference of 2. Understanding how to sum these terms is crucial for proving the statement S_n.
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Mathematical Induction
Mathematical induction is a proof technique used to establish the truth of an infinite sequence of statements. It involves two steps: proving the base case (usually for n=1) and then showing that if the statement holds for n=k, it must also hold for n=k+1. This method is essential for demonstrating that the statement S_n is true for all positive integers n.
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Quadratic Functions
A quadratic function is a polynomial function of degree two, typically expressed in the form f(n) = n^2. In the context of the statement S_n, the right side of the equation represents the square of n, which is significant because it suggests a relationship between the sum of the first n odd numbers and the area of a square with side length n. Recognizing this relationship is key to understanding the proof.
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