Let us assume that there is a positive integer n for which √n-1 + √n+1 is rational and equal to A/B where A and B are positive integersB ≠ 0. Then Since A and B are positive integers.
=> √n-1 and √n+1 are rationals. But it is possible only when n + 1 and n - 1 both are perfect squares. But they differ by 2 and any two perfect squares differ at least by 3.
=> n + 1 and n - 1 cannot be perfect squares. Hence there is no positive integer n for which √n-1 + √n+1 is rational.
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Answer:
Let us assume that there is a positive integer n for which √n-1 + √n+1 is rational and equal to A/B where A and B are positive integersB ≠ 0. Then Since A and B are positive integers.
=> √n-1 and √n+1 are rationals. But it is possible only when n + 1 and n - 1 both are perfect squares. But they differ by 2 and any two perfect squares differ at least by 3.
=> n + 1 and n - 1 cannot be perfect squares. Hence there is no positive integer n for which √n-1 + √n+1 is rational.
Step-by-step explanation:
Let us assume that there is a positive integer n for which √n-1 + √n+1 is rational and equal to A/B where A and B are positive integersB ≠ 0.
Then Since A and B are positive integers. => √n-1 and √n+1 are rationals.
But it is possible only when n + 1 and n - 1 both are perfect squares
HOPE IT HELPS..