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Show that there is no positive integer n for which √n-1 + √n+ 1 is rational?
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Show that there is no positive integer n for which √n-1 + √n+ 1 is rat...
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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.
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Show that there is no positive integer n for which √n-1 + √n+ 1 is rational?
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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?.
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