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Show that there is no positive integer n for which v(n-1) +v(n+1) is rational

Show that there is no positive integer n for which v(n-1) +v(n+1) is rational

Grade:12

1 Answers

Sher Mohammad IIT Delhi
askIITians Faculty 174 Points
9 years ago
sqrt(n-1) + sqrt(n+1) = sqrt(2), for n=1

Suppose 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 that
sqrt(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 mohammad
b.tech, iit delhi

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