Question

Use mathematical induction to prove that each statement is true for every positive integer n. 2 is a factor of n2 - n.

Accepted Answer

First, identify the statement to prove using mathematical induction: For every positive integer $$n$$, $$2 is a $$factor of $$n^{2} - n$. This means we want to show that n$$^{2}$$ - n is $$divisible by $$2$$ for all $$n \geq 1. $$Start with the base case: check if the statement holds for $$n = 1. $$Substitute $$n = 1$$ into the expression $$n^{2} - n$ and verify if the result is divisible by 2$. Assume the statement is true for some positive integer k$, i.e., assume 2$ divides k$$^{2}$$ - k. $$This is called the induction hypothesis. Using the induction hypothesis, prove the statement for $$n = k + 1. $$Substitute $$k + 1$$ into the expression to get $$(k + 1$$)^{2}$$ - (k + 1)$$ and simplify it. Show that the expression for $$n = k + 1$$ can be written as a sum of terms, one of which is divisible by $$2 by $$the induction hypothesis, and the other terms are also divisible by $$2. $$This will complete the induction step and prove the statement for all positive integers $$n.$$

Use mathematical induction to prove that each statement is true for every positive integer n. 2 is a factor of n² - n.

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Key Concepts

Mathematical Induction

Divisibility and Factors

Algebraic Manipulation of Expressions