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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1. Identify the given statement S_n: the sum of the sequence 3 + 7 + 11 + ... + (4n - 1) equals n(2n + 1). This is a formula for the sum of the first n terms of the sequence where the nth term is given by a_n = 4n - 1.
2. Write the statement S_k by replacing n with k: S_k states that 3 + 7 + 11 + ... + (4k - 1) = k(2k + 1). This represents the sum of the first k terms of the sequence.
3. Write the statement S_(k+1) by replacing n with k + 1: S_(k+1) states that 3 + 7 + 11 + ... + (4(k + 1) - 1) = (k + 1)(2(k + 1) + 1). This represents the sum of the first k + 1 terms of the sequence.
4. Simplify the right-hand side of S_(k+1): (k + 1)(2(k + 1) + 1) = (k + 1)(2k + 2 + 1) = (k + 1)(2k + 3).
5. Express S_(k+1) in terms of S_k plus the next term: The sum up to k + 1 terms equals the sum up to k terms plus the (k + 1)th term. So, S_(k+1) = S_k + a_(k+1) = k(2k + 1) + [4(k + 1) - 1].

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite sequence of statements. It involves proving a 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 formulas involving sums or sequences.

Arithmetic Sequences and Series

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. The sum of the first n terms can be expressed using a formula. Understanding how to identify the nth term and sum of an arithmetic series is crucial for simplifying and verifying the given sum expression.

Algebraic Simplification

Algebraic simplification involves manipulating expressions to their simplest form by combining like terms, factoring, or expanding. Simplifying S_(k+1) requires careful algebraic steps to rewrite the sum expression clearly, which helps in comparing it to the proposed formula and completing the induction step.