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Show that for any positive integer n ,(√n 1) (√n-1) is not a rational number?
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Show that for any positive integer n ,(√n 1) (√n-1) is not a rationa...
Let √n−1+√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 0
squaring on both sides
we get n-1+n+1+2 √n2−1
2n+2 √n2−1 =p2q2
2(n+√n2−1)=p2q2
2(n+√n2−1)q2=p2
this mean 2 divides p2 and also divides p.
then let p=2k for any integer k
then 2(n+√n2−1)=(2k)2q2
2(n+√n2−1)=4k2q2
q2=2k2/(n+√n2−1)
so 2 divides q2 and also q
p and q have common factors 2 which contradicts the fact that p and q are co-primes which is due to our wrong assumption. so
√n−1+√n+1 is irrational.
squaring on both sides
we get n-1+n+1+2 √n2−1
2n+2 √n2−1 =p2q2
2(n+√n2−1)=p2q2
2(n+√n2−1)q2=p2
this mean 2 divides p2 and also divides p.
then let p=2k for any integer k
then 2(n+√n2−1)=(2k)2q2
2(n+√n2−1)=4k2q2
q2=2k2/(n+√n2−1)
so 2 divides q2 and also q
p and q have common factors 2 which contradicts the fact that p and q are co-primes which is due to our wrong assumption. so
√n−1+√n+1 is irrational.
Community Answer
Show that for any positive integer n ,(√n 1) (√n-1) is not a rationa...
Proof that (√n + 1) / (√n - 1) is not a rational number:
- Assume (√n + 1) / (√n - 1) is rational:
Assume that (√n + 1) / (√n - 1) is a rational number, which means it can be expressed as a fraction of two integers: a/b, where a and b are integers and b is not equal to 0.
- Express the given expression in terms of a and b:
(√n + 1) / (√n - 1) = a/b
This can be rearranged to: √n + 1 = a/b(√n - 1)
- Simplify the expression:
√n + 1 = (a√n - a) / b
√n + 1 = (a√n - a) / b
√n + 1 = (a√n - a) / b
b(√n + 1) = a√n - a
b√n + b = a√n - a
b + a = √n(a - b)
- Contradiction:
This equation shows that √n is a rational number, which is a contradiction since the square root of a non-perfect square is irrational. Therefore, our initial assumption that (√n + 1) / (√n - 1) is rational is false. Hence, it is proved that (√n + 1) / (√n - 1) is not a rational number.
- Assume (√n + 1) / (√n - 1) is rational:
Assume that (√n + 1) / (√n - 1) is a rational number, which means it can be expressed as a fraction of two integers: a/b, where a and b are integers and b is not equal to 0.
- Express the given expression in terms of a and b:
(√n + 1) / (√n - 1) = a/b
This can be rearranged to: √n + 1 = a/b(√n - 1)
- Simplify the expression:
√n + 1 = (a√n - a) / b
√n + 1 = (a√n - a) / b
√n + 1 = (a√n - a) / b
b(√n + 1) = a√n - a
b√n + b = a√n - a
b + a = √n(a - b)
- Contradiction:
This equation shows that √n is a rational number, which is a contradiction since the square root of a non-perfect square is irrational. Therefore, our initial assumption that (√n + 1) / (√n - 1) is rational is false. Hence, it is proved that (√n + 1) / (√n - 1) is not a rational number.
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Show that for any positive integer n ,(√n 1) (√n-1) is not a rational number? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Show that for any positive integer n ,(√n 1) (√n-1) is not a rational number? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Show that for any positive integer n ,(√n 1) (√n-1) is not a rational number?.
Show that for any positive integer n ,(√n 1) (√n-1) is not a rational number? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Show that for any positive integer n ,(√n 1) (√n-1) is not a rational number? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Show that for any positive integer n ,(√n 1) (√n-1) is not a rational number?.
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