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Question Number 191525 by mehdee42 last updated on 25/Apr/23
Q : Show that the numbers (√(3 )) , 2 & (√8) cannot be terms of an arithmetic sequence.
Q:Showthatthenumbers3,2&8cannotbetermsofanarithmeticsequence.
Answered by mr W last updated on 26/Apr/23
(√3)<2<(√8)  assume they are terms of an A.P.,  then we have  2=(√3)+md  (√8)=2+nd  with m,n ∈N, d ∈R.  d=((2−(√3))/m)=(((√8)−2)/n)  ⇒2(m+n)=m(√8)+n(√3)  ⇒4(m+n)^2 =8m^2 +3n^2 +4mn(√6)  ⇒(√6)=((4(m+n)^2 −8m^2 −3n^2 )/(4mn))=((integer)/(integer))  i.e. (√6) is rational, but (√6) is in fact not  rational.   contradiction!  ⇒(√3), 2, (√8) can not be terms of an A.P.!
3<2<8assumetheyaretermsofanA.P.,thenwehave2=3+md8=2+ndwithm,nN,dR.d=23m=82n2(m+n)=m8+n34(m+n)2=8m2+3n2+4mn66=4(m+n)28m23n24mn=integerintegeri.e.6isrational,but6isinfactnotrational.contradiction!3,2,8cannotbetermsofanA.P.!
Commented by mehdee42 last updated on 27/Apr/23
beravo sir W  the result of  contradiction can also be obtained frome the relation  ((2−(√3))/m)=(((√8)−2)/n)⇒((2−(√3))/( (√8)−2)) =(m/n) ,  because we have rational number on one side anf dumb nimber on thr other side
beravosirWtheresultofcontradictioncanalsobeobtainedfrometherelation23m=82n2382=mn,becausewehaverationalnumberononesideanfdumbnimberonthrotherside
Commented by mehdee42 last updated on 27/Apr/23
anyway , your proof  for this contradiction is appriciated
anyway,yourproofforthiscontradictionisappriciated
Commented by mr W last updated on 27/Apr/23
but we can not generally say that  ((irrational number)/(irrational number)) is also irrational.
butwecannotgenerallysaythatirrationalnumberirrationalnumberisalsoirrational.
Commented by mehdee42 last updated on 28/Apr/23
exactly
exactly

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