show that there is no positive n for which square root of n minus 1 end root +square root of n plus 1 end root  is rational
 

Asked by manavmansvi | 27th Jun, 2015, 06:40: PM

Expert Answer:

A s s u m e space t h a t space square root of n minus 1 end root plus square root of n plus 1 end root i s space r a t i o n a l. rightwards double arrow square root of n minus 1 end root plus square root of n plus 1 end root equals p over q comma space w h e r e space p space a n d space q space a r e space i n t e g e r s comma space q not equal to 0... left parenthesis 1 right parenthesis rightwards double arrow fraction numerator 1 over denominator square root of n minus 1 end root plus square root of n plus 1 end root end fraction equals q over p rightwards double arrow fraction numerator 1 over denominator square root of n minus 1 end root plus square root of n plus 1 end root end fraction cross times fraction numerator square root of n minus 1 end root minus square root of n plus 1 end root over denominator square root of n minus 1 end root minus square root of n plus 1 end root end fraction equals q over p rightwards double arrow fraction numerator square root of n minus 1 end root minus square root of n plus 1 end root over denominator n minus 1 minus open parentheses n plus 1 close parentheses end fraction equals q over p rightwards double arrow fraction numerator square root of n minus 1 end root minus square root of n plus 1 end root over denominator n minus 1 minus n minus 1 end fraction equals q over p rightwards double arrow fraction numerator square root of n minus 1 end root minus square root of n plus 1 end root over denominator negative 2 end fraction equals q over p rightwards double arrow fraction numerator square root of n plus 1 end root minus square root of n minus 1 end root over denominator 2 end fraction equals q over p rightwards double arrow square root of n plus 1 end root minus square root of n minus 1 end root equals fraction numerator 2 q over denominator p end fraction... left parenthesis 2 right parenthesis A d d i n g space e q u a t i o n s space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space open parentheses square root of n minus 1 end root plus square root of n plus 1 end root close parentheses plus open parentheses square root of n plus 1 end root minus square root of n minus 1 end root close parentheses equals p over q plus fraction numerator 2 q over denominator p end fraction rightwards double arrow open parentheses square root of n minus 1 end root plus square root of n plus 1 end root close parentheses plus open parentheses square root of n plus 1 end root minus square root of n minus 1 end root close parentheses equals fraction numerator p squared plus 2 q squared over denominator p q end fraction rightwards double arrow 2 square root of n plus 1 end root equals fraction numerator p squared plus 2 q squared over denominator p q end fraction rightwards double arrow square root of n plus 1 end root equals fraction numerator p squared plus 2 q squared over denominator 2 p q end fraction rightwards double arrow n plus 1 equals open square brackets fraction numerator p squared plus 2 q squared over denominator 2 p q end fraction close square brackets squared... left parenthesis 3 right parenthesis S u b t r a c t i n g space e q u a t i o n s space left parenthesis 2 right parenthesis space f r o m space left parenthesis 1 right parenthesis comma space w e space h a v e comma space open parentheses square root of n minus 1 end root plus square root of n plus 1 end root close parentheses minus open parentheses square root of n plus 1 end root minus square root of n minus 1 end root close parentheses equals p over q minus fraction numerator 2 q over denominator p end fraction rightwards double arrow open parentheses square root of n minus 1 end root plus square root of n plus 1 end root close parentheses minus open parentheses square root of n plus 1 end root minus square root of n minus 1 end root close parentheses equals fraction numerator p squared minus 2 q squared over denominator p q end fraction rightwards double arrow 2 square root of n minus 1 end root equals fraction numerator p squared minus 2 q squared over denominator p q end fraction rightwards double arrow square root of n minus 1 end root equals fraction numerator p squared minus 2 q squared over denominator 2 p q end fraction rightwards double arrow n minus 1 equals open square brackets fraction numerator p squared minus 2 q squared over denominator 2 p q end fraction close square brackets squared... left parenthesis 4 right parenthesis A d d i n g space e q u a t i o n s space left parenthesis 3 right parenthesis space a n d space left parenthesis 4 right parenthesis comma space w e space h a v e comma n plus 1 plus n minus 1 equals open square brackets fraction numerator p squared plus 2 q squared over denominator 2 p q end fraction close square brackets squared minus open square brackets fraction numerator p squared minus 2 q squared over denominator 2 p q end fraction close square brackets squared rightwards double arrow n plus 1 plus n minus 1 equals 2... left parenthesis 5 right parenthesis F r o m space e q u a t i o n s space left parenthesis 3 right parenthesis space a n d space left parenthesis 4 right parenthesis comma space i t space i s space c l e a r space t h a t space n plus 1 space a n d space n minus 1 space a r e space p e r f e c t space s q u a r e s. B u t space a n y space t w o space p e r f e c t space s q u a r e s space d i f f e r space b y space a t l e a s t space 3. T h u s comma space o u r space a s s u m p t i o n space t h a t space square root of n minus 1 end root plus square root of n plus 1 end root i s space r a t i o n a l space i s space i n c o r r e c t. H e n c e comma space t h e r e space i s space n o space p o s i t i v e space apostrophe n apostrophe space s u c h space t h a t space square root of n minus 1 end root plus square root of n plus 1 end root i s space r a t i o n a l.

Answered by Vimala Ramamurthy | 28th Jun, 2015, 12:29: PM

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