Sunday, 18 March 2018

algebra precalculus - Why k should be odd?




My teacher once said, for any positive number  n,  nk1 would always have  n1 as a factor for all positive odd values of  k. Could anyone tell me the proof? I have written my approach below.



Assuming n1 to be a factor.



nk1=(n1)x



nk=(n1)x+1




k=lg((n1)x+1)/lg(n)



If my approach is right could you tell me where should I go from here?



Edit : This clears my doubt.



Edit 2 : My question is "why for any positive number  n,  nk+1 would always have  n+1 as a factor for all positive odd values of  k ".


Answer



More generally, if a,bZ, nZ+, then abanbn.




It follows from anbn=(ab)(an1+an2b++bn1)



Edit: Since it seems you wanted to ask about the fact that for nZ with odd kZ+ we have n+1nk+1:



More generally, for a,bZ and odd nZ+ we have an+bn=(a+b)×



×(an1an2b+an3b2abn2+bn1)


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