A statement S_n 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 given statement \(S_n\): the sum of the first \(n\) terms of the sequence \(4, 8, 12, \ldots, 4n\) is given by the formula \(4 + 8 + 12 + \cdots + 4n = 2n(n + 1)\).
2. Write the statement \(S_k\) by replacing \(n\) with \(k\): \(4 + 8 + 12 + \cdots + 4k = 2k(k + 1)\).
3. Write the statement \(S_{k+1}\) by replacing \(n\) with \(k+1\): \(4 + 8 + 12 + \cdots + 4k + 4(k+1) = 2(k+1)((k+1) + 1)\).
4. Simplify the right-hand side of \(S_{k+1}\): \$2(k+1)(k+2)$.
5. Express \(S_{k+1}\) as the sum of \(S_k\) plus the next term: \(S_k + 4(k+1) = 2(k+1)(k+2)\), which shows how the formula extends from \(k\) to \(k+1\).

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.

Summation of Arithmetic Sequences

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. The sum of the first n terms can be found using formulas or by recognizing patterns. In this problem, the sequence 4, 8, 12, ... is arithmetic with a common difference of 4.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and applying arithmetic operations to rewrite expressions in a simpler or more standard form. This skill is essential when expressing S_k and S_{k+1} clearly and verifying the induction step.