Here are the essential concepts you must grasp in order to answer the question correctly.
Mathematical Induction
Mathematical induction is a proof technique used to establish the truth of an infinite sequence of statements, typically involving integers. It consists of two main steps: the base case, where the statement is verified for the initial value (usually n=1), and the inductive step, where one assumes the statement holds for n=k and then proves it for n=k+1. This method is essential for proving statements that are asserted for all positive integers.
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Arithmetic Series
An arithmetic series is the sum of the terms of an arithmetic sequence, where each term after the first is obtained by adding a constant difference. In this case, the series 5 + 10 + 15 + ... + 5n can be expressed as 5 times the sum of the first n positive integers. Understanding the formula for the sum of the first n integers, which is n(n+1)/2, is crucial for simplifying and proving the given statement.
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Formula for the Sum of an Arithmetic Series
The formula for the sum of an arithmetic series can be expressed as S_n = n/2 * (a + l), where S_n is the sum of the first n terms, a is the first term, and l is the last term. For the series in question, the first term is 5, and the last term is 5n. This formula helps in deriving the closed form of the series, which is necessary for the inductive proof to show that the left-hand side equals the right-hand side of the equation.
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