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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 rationa...
letsqrt(n-1)+sqrt(n+1)be a rational number which can be expressed as p/q, p and q are integers and coprime. q is not equal to 0squaring on both sides we get n-1+n+1+2sqrt(n^2-1)2n+2sqrt(n^2-1)=p^2/q^22(n+sqrt(n^2-1))=p^2/q^22(n+sqrt(n^2-1))q^2=p^2this mean 2 dividesp^2and also divides p.then let p=2k for any integer kthen 2(n+sqrt(n^2-1))=(2k)^2/q^22(n+sqrt(n^2-1))=4k^2/q^2q^2=2k^2/(n+sqrt(n^2-1))so 2 dividesq^2and also qp and q have common factors 2 which contradicts the fact that p and q are co-primes which is due to our wrong assumption. sosqrt(n-1)+sqrt(n+1)is irrational.
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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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