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: 4 + 8 + 12 + ... + 4n = 2n(n + 1)

Verified step by step guidance

1. Identify the pattern in the sequence: The sequence given is an arithmetic sequence where each term increases by 4. The nth term can be expressed as 4n.
2. Statement S_k: Substitute k for n in the given statement Sn. This gives us the sum of the first k terms: 4 + 8 + 12 + ... + 4k = 2k(k + 1).
3. Statement S_(k+1): Substitute (k+1) for n in the given statement Sn. This gives us the sum of the first (k+1) terms: 4 + 8 + 12 + ... + 4(k+1) = 2(k+1)((k+1) + 1).
4. Simplify S_(k+1): Expand the expression 2(k+1)((k+1) + 1) to get 2(k+1)(k+2).
5. Verify the relationship between S_k and S_(k+1): Notice that S_(k+1) can be expressed as S_k + 4(k+1), which confirms the arithmetic sequence pattern.

Key Concepts

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence, where each term increases by a constant difference. In this case, the series Sn represents the sum of the first n terms of the sequence 4, 8, 12, ..., which can be expressed as 4 times the sum of the first n positive integers.

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 an arbitrary integer k, it also holds for k+1. This method is essential for validating the formula given in Sn.

Simplifying Expressions

Simplifying expressions involves rewriting mathematical statements in a more concise or manageable form. In the context of this problem, simplifying S_(k+1) means expressing the sum of the series for n=k+1 in terms of n and then reducing it to match the form of the right side of the equation, 2n(n + 1).