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 \(3 + 7 + 11 + \ldots + (4n - 1)\) equals \(n(2n + 1)\).
2. Write the statement \(S_k\) by replacing \(n\) with \(k\): \(3 + 7 + 11 + \ldots + (4k - 1) = k(2k + 1)\).
3. Write the statement \(S_{k+1}\) by replacing \(n\) with \(k+1\): \(3 + 7 + 11 + \ldots + (4k - 1) + [4(k+1) - 1] = (k+1)(2(k+1) + 1)\).
4. Simplify the term \$4(k+1) - 1\( inside the sum: \)4k + 4 - 1 = 4k + 3$.
5. Simplify the right side expression \((k+1)(2(k+1) + 1)\) by expanding the parentheses: \((k+1)(2k + 2 + 1) = (k+1)(2k + 3)\).

Key Concepts

Mathematical Induction

Mathematical induction is a proof technique used to verify statements about positive integers. It involves two steps: proving the base case (usually for n=1) and then proving that if the statement holds for n=k, it also holds for n=k+1. This method confirms the statement for all positive integers.

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 forms an arithmetic series. Understanding how to express and simplify these sums is essential for working with series like 3 + 7 + 11 + ... + (4n - 1).

Algebraic Simplification

Algebraic simplification involves rewriting expressions in simpler or more compact forms by combining like terms, factoring, or expanding. Simplifying S_{k+1} completely requires careful manipulation of algebraic expressions to clearly show the relationship between terms.