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Question Number 192425 by mehdee42 last updated on 17/May/23
Question  if  “k” is odd  & A=1^k +2^k +...+n^(k  )  &  B=1+2+...+n  prove that  :  B ∣ A
Questionifkisodd&A=1k+2k++nk&B=1+2++nprovethat:BA
Answered by MM42 last updated on 24/May/23
We khnow  ;  1+2+...+n=((n(n+1))/2)   &   (n,((n+1)/2))=1 or ((n/2),n+1)=1  Thus ,it soffices  to  show that   n∣A & ((n+1)/2) ∣A  or  (n/2)∣A & n+1∣A  proof ∴  if  “n” is even ⇒   (n/2)∣1^k +(n−1)^k  , 2^k +(n−2)^k  , ... (i)   n+1∣1^k +n^k  , 2^k +(n−1)^k  , ... (ii)  (i),(ii)⇒B∣A  if  “n” is  odd⇒   n∣1^k +(n−1)^k  , 2^k +(n−2)^k  , ... (iii)   ((n+1)/2)∣1^k +n^k  , 2^k +(n−1)^k  , ... (iv)  (iii),(iv)⇒B∣A  ⇒∀ n∈N ⇒1+2+3+...+n∣ 1^k +2^k +...+n^k    ; if  “k” is  “odd”
Wekhnow;1+2++n=n(n+1)2&(n,n+12)=1or(n2,n+1)=1Thus,itsofficestoshowthatnA&n+12Aorn2A&n+1Aproofifnisevenn21k+(n1)k,2k+(n2)k,(i)n+11k+nk,2k+(n1)k,(ii)(i),(ii)BAifnisoddn1k+(n1)k,2k+(n2)k,(iii)n+121k+nk,2k+(n1)k,(iv)(iii),(iv)BAnN1+2+3++n1k+2k++nk;ifkisodd

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