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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Geometric Series
A geometric series is a series of terms where each term after the first is found by multiplying the previous term by a constant called the common ratio. In the given statement, the series 1 + 4 + 4^2 + ... + 4^(n-1) is a geometric series with a first term of 1 and a common ratio of 4. Understanding the formula for the sum of a geometric series is crucial for simplifying and proving the statement.
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Formula for the Sum of a Geometric Series
The formula for the sum of the first n terms of a geometric series is S_n = a(1 - r^n) / (1 - r), where 'a' is the first term and 'r' is the common ratio. For the series in the question, this formula can be applied to derive the expression ((4^n) - 1) / 3. Recognizing how to manipulate this formula is key to completing the proof using mathematical induction.
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