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)
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Mathematical Induction
Mathematical induction is a proof technique used to establish the truth of an infinite sequence of statements. It consists of two main steps: the base case, where the statement is verified for the initial value (usually n=1), and the inductive step, where one assumes the statement holds for n=k and then proves it for n=k+1. This method is particularly useful for proving formulas involving integers.
The expression 4 + 8 + 12 + ... + 4n represents an arithmetic series where each term increases by a constant difference. The sum of the first n terms of an arithmetic series can be calculated using the formula S_n = n/2 * (first term + last term). Understanding how to derive and manipulate this formula is essential for proving the given statement using induction.
The right side of the equation, 2n(n + 1), is a quadratic function in terms of n. Quadratic functions are polynomials of degree two and can be expressed in the standard form ax^2 + bx + c. Recognizing the properties of quadratic functions, such as their growth rate and the ability to factor them, is crucial for verifying the equality in the induction proof.