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Show that there is no positive integer n for which root n-1 + root n+1 is rational
sqrt(n-1) + sqrt(n+1) = sqrt(2), for n=1Suppose that n > 1.Suppose that sqrt(n-1) + sqrt(n+1) is rational.Then its square [sqrt(n-1) + sqrt(n+1)]^2 = 2n + 2 * sqrt(n^2 -1) is also rational.Next, since 2 and n are rational, by the closure laws of Q, we have thatsqrt(n^2 - 1) is rational. This proof will be complete if we can prove the following fact.Claim: sqrt(n^2 - 1) is irrational.This follows from the claim that consecutive squares are spaced more than 1 unit apart as long as n^2 > 1. [(n+1)^2 - n^2 = 2n + 1.]More precisely, since (n - 1)^2 < n^2 - 1< n^2 for all integers n > 1, taking square roots shows that sqrt(n^2 - 1) is between two consecutive perfect squares.sher mohammadb.tech, iit delhi
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