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Show that there is no positive integer n for which √(n-1) +√(n+1) is rational.{In this question I am confused about the contradiction that any two perfect squares differ at least by 3. But some numbers like √2.25 & √4 have 1.5&2 whose difference is 0.5 . So how is it possible?}
Asked by Ananay | 05 Apr, 2017, 06:14: PM
answered-by-expert Expert Answer

begin mathsize 16px style square root of left parenthesis straight n minus 1 right parenthesis end root space plus square root of left parenthesis straight n plus 1 right parenthesis end root space suppose space it space is space rational. square root of left parenthesis straight n minus 1 right parenthesis end root space plus square root of left parenthesis straight n plus 1 right parenthesis end root space equals straight m over straight n straight m space and space straight n space are space positive comma space straight n space not space equal space to space zero comma straight n over straight m equals fraction numerator 1 over denominator square root of left parenthesis straight n minus 1 right parenthesis end root space plus square root of left parenthesis straight n plus 1 right parenthesis end root end fraction After space doing space rationalisation space you space will space get comma square root of left parenthesis straight n minus 1 right parenthesis end root equals fraction numerator straight m squared minus 2 straight n squared over denominator 2 mn end fraction Hence comma space square root of left parenthesis straight n minus 1 right parenthesis end root space and space square root of left parenthesis straight n plus 1 right parenthesis end root space are space rational. end style

Answered by | 05 Apr, 2017, 06:25: PM

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