In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n. 4 + 8 + 12 + ... + 4n = 2n(n + 1)

Verified step by step guidance

1. Step 1: Understand the statement to prove by induction. We want to prove that for every positive integer $n$, the sum $4\; +\; 8\; +\; 12\; +\; \backslash cdots\; +\; 4n$ equals $2n(n\; +\; 1)$.
2. Step 2: Base Case - Verify the statement for $n\; =\; 1$. Substitute $n\; =\; 1$ into both sides: Left side is $4\; \backslash times\; 1\; =\; 4$, and right side is $2\; \backslash times\; 1\; \backslash times\; (1\; +\; 1)\; =\; 2\; \backslash times\; 1\; \backslash times\; 2\; =\; 4$. Since both sides are equal, the base case holds.
3. Step 3: Inductive Hypothesis - Assume the statement is true for some positive integer $k$, that is, assume $4\; +\; 8\; +\; 12\; +\; \backslash cdots\; +\; 4k\; =\; 2k(k\; +\; 1)$.
4. Step 4: Inductive Step - Prove the statement for $k\; +\; 1$. Start with the sum up to $k\; +\; 1$: $4\; +\; 8\; +\; 12\; +\; \backslash cdots\; +\; 4k\; +\; 4(k\; +\; 1)$. Using the inductive hypothesis, replace the sum up to $k$ with $2k(k\; +\; 1)$, so the sum becomes $2k(k\; +\; 1)\; +\; 4(k\; +\; 1)$.
5. Step 5: Simplify the expression $2k(k\; +\; 1)\; +\; 4(k\; +\; 1)$ by factoring out $(k\; +\; 1)$: $(k\; +\; 1)(2k\; +\; 4)$. Then factor further to get $2(k\; +\; 1)(k\; +\; 2)$, which matches the right side of the statement for $n\; =\; k\; +\; 1$. This completes the inductive step.

Key Concepts

Mathematical Induction

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

Arithmetic Series

An arithmetic series is the sum of terms in an arithmetic sequence, where each term increases by a constant difference. In this problem, the series 4 + 8 + 12 + ... + 4n has a common difference of 4. Understanding how to sum such series is key to verifying the formula.

Algebraic Manipulation

Algebraic manipulation involves simplifying expressions and equations using algebraic rules. To prove the formula, one must manipulate sums and expressions, such as factoring and expanding, to show equivalence between the series sum and the given formula 2n(n + 1).