Proving ( n^2 - n ) is Divisible by 2 Using Mathematical Induction

Mathematical induction is a powerful tool in proving statements about natural numbers. In this article, we will use mathematical induction to prove that ( n^2 - n ) is divisible by 2 for all natural numbers ( n geq 1 ). This proof follows a structured pattern: we first establish the base case, then assume the statement is true for some arbitrary ( n k ), and finally prove it is true for ( n k 1 ). Here is a detailed step-by-step breakdown of the proof process.

Step 1: Base Case

The base case is to verify the statement for ( n 1 ). For ( n 1 ): ( 1^2 - 1 0 ) 0 is divisible by 2. This establishes the base case.

Step 2: Inductive Hypothesis

Assume that the statement is true for ( n k ) where ( k ) is any positive integer. That is, assume ( k^2 - k ) is divisible by 2. Mathematically, we can write this as:

k^2 - k 2m quad text{where} ; m , text{is an integer}

Step 3: Inductive Step

We need to show that ( (k 1)^2 - (k 1) ) is divisible by 2. Let's start with the expression: ( (k 1)^2 - (k 1) ) ( (k 1)^2 - (k 1) (k^2 2k 1) - (k 1) ) ( k^2 2k 1 - k - 1 k^2 k ) ( (k^2 - k) 2k ) Now, substituting the inductive hypothesis ( k^2 - k 2m ), we get: ( (k^2 - k) 2k 2m 2k ) ( 2(m k) ) ( 2 times text{an integer} ) Since ( m k ) is an integer, ( 2(m k) ) is clearly divisible by 2. This completes the inductive step.

Conclusion

By mathematical induction, we have shown that ( n^2 - n ) is divisible by 2 for all natural numbers ( n ).

Alternative Proof: Product of Consecutive Integers

Another approach to proving this statement is by recognizing the property of the product of consecutive integers. The expression ( n^2 - n ) can be rewritten as ( n(n - 1) ), which is the product of two consecutive integers. One of these integers must be even, and an even number is always divisible by 2. Therefore, the product of two consecutive integers is always divisible by 2.

Summary

In this article, we have presented two different methods to prove that ( n^2 - n ) is divisible by 2 using mathematical induction and the property of consecutive integers. Both methods confirm that the statement holds true for all natural numbers ( n ). This proof provides a foundational understanding of mathematical induction and its applications in number theory.