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 number of statements, typically involving positive 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 defined recursively or involve sequences.
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Properties of Exponents
The properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the power of a product ( (ab)^n = a^n * b^n). Understanding these properties is crucial for simplifying expressions and proving identities involving exponents.
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Base Case and Inductive Step
In mathematical induction, the base case is the initial step that verifies the statement for the smallest integer, usually n=1. The inductive step involves assuming the statement is true for an arbitrary positive integer n=k and then demonstrating that it must also be true for n=k+1. This two-part structure is fundamental to the induction process, ensuring that if the base case holds, the truth of the statement can be extended to all positive integers.
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