Use mathematical induction to prove that each statement is true for every positive integer n. 3

step 1 So we want to use induction to prove that 3 is a factor of n times n plus 1 times n minus 1 for

Use mathematical induction to prove that each statement is...

Key Concepts

Consecutive Integers

Consecutive integers are numbers that follow each other in order without gaps, such as n-1, n, and n+1. An important property of consecutive integers is that they cover all remainders when divided by any number. For instance, among any three consecutive integers, one must be divisible by 3, which is a key observation in proving divisibility results in many induction proofs.

Divisibility

Divisibility is a fundamental concept in number theory that involves determining whether one integer is a factor of another. In proofs involving divisibility, algebraic manipulation and properties of integers are used to show that a given product or sum has a specific divisor. This concept is central when proving that certain expressions are divisible by numbers like 3, regardless of the value of n.

Inductive Step

The inductive step is the phase of an induction proof where, under the assumption that the statement is true for an arbitrary integer k, you prove that the statement must also hold for k+1. Successfully completing this step, together with the base case, establishes the statement for all positive integers.

Base Case

The base case is the first step in an induction proof where you check that the statement holds for the initial value, typically n = 1 (or another specified starting point). This step is crucial because it serves as the foundation that the rest of the proof builds upon.

Mathematical Induction

Mathematical induction is a proof technique used to establish that a given statement holds true for all positive integers. The process involves proving a base case, where the statement is verified for an initial value, and then proving an inductive step, where one shows that if the statement is true for an arbitrary integer k, it must also be true for k+1.

Inductive Hypothesis

The inductive hypothesis involves assuming that the statement holds true for some arbitrary positive integer k. This assumption is used as a stepping stone to prove that the statement must then hold for the next integer, k+1, thereby linking the truth of the statement from one integer to the next.

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