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 given statement S_n: the sum of the first n terms of the sequence 4, 8, 12, ..., 4n is given by $S{n}_{}=\; 4\; +\; 8\; +\; 12\; +\; \backslash cdots\; +\; 4n\; =\; 2n(n\; +\; 1)$.
2. Write the statement S_k by replacing n with k: $S{k}_{}=\; 4\; +\; 8\; +\; 12\; +\; \backslash cdots\; +\; 4k\; =\; 2k(k\; +\; 1)$.
3. Write the statement S_(k+1) by replacing n with k + 1: $S{\mathrm{k+1}}_{}=\; 4\; +\; 8\; +\; 12\; +\; \backslash cdots\; +\; 4k\; +\; 4(k\; +\; 1)$.
4. Express S_(k+1) in terms of S_k plus the next term: $S{\mathrm{k+1}}_{}=S{k}_{}+\; 4(k\; +\; 1)$.
5. Substitute the expression for S_k into the equation and simplify: $S{\mathrm{k+1}}_{}=\; 2k(k\; +\; 1)\; +\; 4(k\; +\; 1)\; =\; 2(k\; +\; 1)(k\; +\; 2)$.

Key Concepts

Mathematical Induction

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

Summation of Arithmetic Series

An arithmetic series is the sum of terms in an arithmetic sequence, where each term increases by a constant difference. The given series 4 + 8 + 12 + ... + 4n is arithmetic with first term 4 and common difference 4. Understanding how to express and simplify such sums is essential for verifying the formula.

Algebraic Simplification

Algebraic simplification involves rewriting expressions in simpler or more compact forms by combining like terms and factoring. Simplifying S_(k+1) requires substituting and manipulating expressions to verify the formula, ensuring clarity and correctness in the induction step.