Class 10 Exam > Class 10 Questions > Show that there is no positive integer n for ...
Start Learning for Free
Show that there is no positive integer n for which √n-1 + √n+ 1 is rational?
Most Upvoted Answer
Show that there is no positive integer n for which √n-1 + √n+ 1 is rat...
Community Answer
Show that there is no positive integer n for which √n-1 + √n+ 1 is rat...
Proof that there is no positive integer n for which √n-1 √n 1 is rational
Assumption
We assume that there exists a positive integer n such that √n-1 √n 1 is rational.
Deriving a Contradiction
Let's simplify the expression by multiplying ( √n-1 + 1) with (√n-1 - 1):
(√n-1 + 1) (√n-1 - 1) = (√n-1)² - 1
= n - 2√n + 1 - 1
= n - 2√n
We know that √n is irrational if n is not a perfect square. So, if √n is irrational, then n - 2√n is also irrational.
If √n is rational, then we can express it as a fraction p/q, where p and q are coprime integers. Substituting this value in n - 2√n, we get:
n - 2√n = n - 2(p/q) = (nq - 2p)/q
Since p and q are coprime, nq - 2p and q are also coprime. Hence, n - 2√n is irrational.
Thus, we have derived a contradiction. Hence, our assumption that there exists a positive integer n such that √n-1 √n 1 is rational is false.
Conclusion
Therefore, there is no positive integer n for which √n-1 √n 1 is rational.
Attention Class 10 Students!
To make sure you are not studying endlessly, EduRev has designed Class 10 study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Class 10.
|
Explore Courses for Class 10 exam
|
|
Similar Class 10 Doubts
Top Courses for Class 10View all
Show that there is no positive integer n for which √n-1 + √n+ 1 is rational?
Question Description
Show that there is no positive integer n for which √n-1 + √n+ 1 is rational? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Show that there is no positive integer n for which √n-1 + √n+ 1 is rational? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Show that there is no positive integer n for which √n-1 + √n+ 1 is rational?.
Show that there is no positive integer n for which √n-1 + √n+ 1 is rational? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Show that there is no positive integer n for which √n-1 + √n+ 1 is rational? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Show that there is no positive integer n for which √n-1 + √n+ 1 is rational?.
Solutions for Show that there is no positive integer n for which √n-1 + √n+ 1 is rational? in English & in Hindi are available as part of our courses for Class 10.
Download more important topics, notes, lectures and mock test series for Class 10 Exam by signing up for free.
Here you can find the meaning of Show that there is no positive integer n for which √n-1 + √n+ 1 is rational? defined & explained in the simplest way possible. Besides giving the explanation of
Show that there is no positive integer n for which √n-1 + √n+ 1 is rational?, a detailed solution for Show that there is no positive integer n for which √n-1 + √n+ 1 is rational? has been provided alongside types of Show that there is no positive integer n for which √n-1 + √n+ 1 is rational? theory, EduRev gives you an
ample number of questions to practice Show that there is no positive integer n for which √n-1 + √n+ 1 is rational? tests, examples and also practice Class 10 tests.
|
|
Explore Courses for Class 10 exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup