In Exercises 11–24, use mathematical induction to prove that each statement is true for every positive integer n. 3 + 7 + 11 + ... + (4n - 1) = n(2n + 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 $3\; +\; 7\; +\; 11\; +\; \backslash cdots\; +\; (4n\; -\; 1)$ equals $n(2n\; +\; 1)$.
2. Step 2: Base Case: Verify the statement for $n\; =\; 1$. Substitute $n\; =\; 1$ into both sides of the equation and check if they are equal. The left side is just the first term $4(1)\; -\; 1\; =\; 3$, and the right side is $1(2(1)\; +\; 1)\; =\; 3$.
3. Step 3: Inductive Hypothesis: Assume the statement is true for some positive integer $k$, that is, assume $3\; +\; 7\; +\; 11\; +\; \backslash cdots\; +\; (4k\; -\; 1)\; =\; k(2k\; +\; 1)$ holds.
4. Step 4: Inductive Step: Prove the statement for $k\; +\; 1$. Start with the sum up to $k\; +\; 1$ terms: $3\; +\; 7\; +\; 11\; +\; \backslash cdots\; +\; (4k\; -\; 1)\; +\; (4(k\; +\; 1)\; -\; 1)$. Use the inductive hypothesis to replace the sum up to $k$ terms, then simplify the expression to show it equals $(k\; +\; 1)(2(k\; +\; 1)\; +\; 1)$.
5. Step 5: Conclude that since the base case is true and the inductive step holds, by mathematical induction, the formula $3\; +\; 7\; +\; 11\; +\; \backslash cdots\; +\; (4n\; -\; 1)\; =\; n(2n\; +\; 1)$ is true for all positive integers $n$.

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 sequence 3, 7, 11, ... has a common difference of 4. Understanding how to express and sum such sequences is essential for formulating and proving the given formula.

Formula for the nth Term of an Arithmetic Sequence

The nth term of an arithmetic sequence is given by a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference. Here, the nth term is expressed as 4n - 1, which helps identify the sequence's pattern and is crucial for setting up the induction proof.