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)

Verified step by step guidance

1. Identify the pattern in the sequence: The sequence given is an arithmetic sequence where each term is of the form \(4n - 1\).
2. Write the statement \(S_k\): This is the statement for the positive integer \(k\), which is \(3 + 7 + 11 + \ldots + (4k - 1) = k(2k + 1)\).
3. Write the statement \(S_{k+1}\): This is the statement for the positive integer \(k+1\), which is \(3 + 7 + 11 + \ldots + (4k - 1) + (4(k+1) - 1) = (k+1)(2(k+1) + 1)\).
4. Simplify \(S_{k+1}\): Simplify the right-hand side of the equation \((k+1)(2(k+1) + 1)\) to get \((k+1)(2k + 2 + 1) = (k+1)(2k + 3)\).
5. Verify the simplification: Ensure that the left-hand side \(3 + 7 + 11 + \ldots + (4k - 1) + (4(k+1) - 1)\) matches the simplified right-hand side \((k+1)(2k + 3)\).

Key Concepts

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

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

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.