In Exercises 5–10, a statement Sn about the positive integers is given. Write statements S_k and S_(k+1) simplifying statement S_(k+1) completely.
Sn: 3 + 7 + 11 + ... + (4n - 1) = n(2n + 1)
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Key Concepts
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 after the first is obtained by adding a constant difference. In this case, the series Sn consists of terms that increase by 4, starting from 3. Understanding how to derive the sum of such sequences is crucial for simplifying the given statement.
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 (usually for n=1) and then showing that if the statement holds for n=k, it also holds for n=k+1. This method is essential for validating the formula provided in Sn.
Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. In the context of this problem, simplifying S_(k+1) requires applying these rules to express the sum in a more manageable form, which is necessary for proving the equality stated in Sn.