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. Identify the statement to prove using mathematical induction: For every positive integer \(n\), the sum \(3 + 7 + 11 + \dots + (4n - 1)\) equals \(n(2n + 1)\).
2. Start with the base case: Verify the statement for \(n = 1\). Substitute \(n = 1\) into both sides of the equation and check if they are equal.
3. Assume the statement is true for some positive integer \(k\), that is, assume \(3 + 7 + 11 + \dots + (4k - 1) = k(2k + 1)\) holds. This is the induction hypothesis.
4. Use the induction hypothesis to prove the statement for \(n = k + 1\). Write the sum up to \(k + 1\) terms as the sum up to \(k\) terms plus the next term: \(3 + 7 + 11 + \dots + (4k - 1) + [4(k + 1) - 1]\).
5. Substitute the induction hypothesis into this expression and simplify algebraically to show that the sum equals \((k + 1)(2(k + 1) + 1)\), completing the induction 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 arbitrary integer k, it also holds for k+1. This establishes the statement for all positive integers.

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 to verify the given formula.

Algebraic Manipulation

Algebraic manipulation involves simplifying expressions and equations using algebraic rules. For this problem, it includes substituting the formula, expanding terms, and rearranging expressions to show equivalence between the sum of the series and the given formula n(2n + 1).