Click to Chat
1800-1023-196
+91-120-4616500
CART 0
Use Coupon: CART20 and get 20% off on all online Study Material
Welcome User
OR
LOGIN
Complete Your Registration (Step 2 of 2 )
Show that there is no positive integer n for which v(n-1) +v(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
Post Question
Dear , Preparing for entrance exams? Register yourself for the free demo class from askiitians.
points won -