Abstract Algebra Theory and Applications - Computer Science ...

24 CHAPTER 1 THE INTEGERSwhere a and b are real numbers, n ∈ N, and( nk)=n!k!(n − k)!is the binomial coefficient. We first show that( ) ( ) ( n + 1 n n= +k k k − 1).This result follows from( ) ( )n n+k k − 1===n!k!(n − k)! + n!(k − 1)!(n − k + 1)!(n + 1)!k!(n + 1 − k)!( ) n + 1.kIf n = 1, the binomial theorem is easy to verify. Now assume that the resultis true for n greater than or equal to 1. Then(a + b) n+1 = (a + b)(a + b) n( n∑ ( n= (a + b)kk=0)a k b n−k )=n∑( nkk=0= a n+1 ++)a k+1 b n−k +n∑( nk − 1k=1n∑( nkk=1= a n+1 +=n∑k=1n+1∑( n + 1kk=0n∑( nkk=0)a k b n+1−k)a k b n+1−k + b n+1[( nk − 1)a k b n+1−k .) ( n+k)a k b n+1−k)]a k b n+1−k + b n+1