Unformatted text preview: , . . . , k , where k ≥ 1. We want to show that x k +1 + 1 /x k +1 is an integer. We have ( x k +1 + 1 /x k +1 ) = ( x k + 1 /x k )( x + 1 /x )( x k1 + 1 /x k1 ) Now, the right hand side is an integer since ( x k + 1 /x k ) , ( x k1 + 1 /x k1 ), and ( x + 1 /x ) are all integers, the ﬁrst two by inductive hypothesis, the later by the original problem’s hypothesis. Hence the left hand side is an integer. By induction, x n + 1 /x n is an integer for all n ∈ N . ± Date : February 21th, 2008. 1...
View
Full Document
 Spring '07
 COURTNEY
 Division, Integers, Mathematical Induction, Inductive Reasoning, Natural number, Lefthandedness, Structural induction, inductive hypothesis

Click to edit the document details