Prove that there is no positive integer n for which n - 1 + n + 1 is rational.

Suppose there exists a positive integer n for which n - 1 + n + 1 is a rational number. Let p q = n - 1 + n + 1 , where p and q are positive integers and q ≠ 0 ⇒ q p = 1 n - 1 + n + 1 ⇒ q p = 1 n - 1 + n + 1 × n - 1 - n + 1 n - 1 - n + 1 = n - 1 - n + 1 ( n - 1 ) - ( n + 1 ) = n - 1 - n + 1 - 2 ⇒ 2 q p = n + 1 - n - 1 Thus, we have, ( n - 1 + n + 1 ) + ( n + 1 - n - 1 ) = p q + 2 q p ⇒ 2 n + 1 = p 2 + 2 q 2 p q ⇒ n + 1 = p 2 + 2 q 2 2 p q - - - - i Also, we have, ( n - 1 + n + 1 ) - ( n + 1 - n - 1 ) = p q - 2 q p ⇒ 2 n - 1 = p 2 - 2 q 2 p q ⇒ n - 1 = p 2 - 2 q 2 2 p q - - - - i i From i and i i , we get, n + 1 and n - 1 are rational. [Since p , q are integers, p 2 + 2 q 2 2 p q and p 2 - 2 q 2 2 p q are rationals]. From i and i i , we also get, n + 1 and n - 1 are perfect square of positive integers. Now, ( n + 1 ) - ( n - 1 ) = 2 , which is not possible since any two perfect squares differ by at least 3 . Hence, there is no positive integer n for which n - 1 + n + 1 is a rational number. [proved]

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Mathematics Crash Course (Based on Revised Syllabus - 2023)>Chapter 1 - Real Numbers>Exercise

Exercise

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