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Trouver-la-somme-de-S-n-Si-S-n-1-11-111-1111-111-




Question Number 175407 by Stephan last updated on 29/Aug/22
               Trouver la somme de S_n        Si      S_n =1+11+111+...+1111...111
TrouverlasommedeSnSiSn=1+11+111++1111111
Answered by Rasheed.Sindhi last updated on 29/Aug/22
   S_n =1+11+111+...+1111...11  9S_n =9+99+999+...+999...99        =(10−1)+(100−1)+(1000−1)+... n terms       =(10+100+1000+...n terms)−n       =((10(10^n −1))/(10−1))−n           =((10(10^n −1)−9n)/9)  S_n =((10^(n+1) −9n−10)/(81))
Sn=1+11+111++1111119Sn=9+99+999++99999=(101)+(1001)+(10001)+nterms=(10+100+1000+nterms)n=10(10n1)101n=10(10n1)9n9Sn=10n+19n1081
Answered by Ar Brandon last updated on 29/Aug/22
u_n =1+11+111+1111+...  ⇒u_n =((10^n −1)/9)  ⇒Σu_n =((10)/9)(((10^n −1)/9))−(n/9)
un=1+11+111+1111+un=10n19Σun=109(10n19)n9

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