Use Coupon: CART20 and get 20% off on all online Study Material

Total Price: Rs.

There are no items in this cart.
Continue Shopping
Grade: 12
        Show that there is no positive integer n for which v(n-1) +v(n+1) is rational
6 years ago

Answers : (1)

Sher Mohammad
IIT Delhi
askIITians Faculty
174 Points
							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, iit delhi
6 years ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies

Course Features

  • 728 Video Lectures
  • Revision Notes
  • Previous Year Papers
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Test paper with Video Solution

Course Features

  • 731 Video Lectures
  • Revision Notes
  • Test paper with Video Solution
  • Mind Map
  • Study Planner
  • NCERT Solutions
  • Discussion Forum
  • Previous Year Exam Questions

Ask Experts

Have any Question? Ask Experts

Post Question

Answer ‘n’ Earn
Attractive Gift
To Win!!! Click Here for details