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Question Number 175407 by Stephan last updated on 29/Aug/22

$$$$$$\underset{―}{\mathrm{Trouver}\mathrm{la}\mathrm{somme}\mathrm{de}{\mathrm{S}}_n}$$$$\mathrm{Si}{S}_n=1+11+111+...+1111...111$$$$$$

Answered by Rasheed.Sindhi last updated on 29/Aug/22

$$S_n=1+11+111+...+1111...11$$$$9S_n=9+99+999+...+999...99$$$$=(10-1)+(100-1)+(1000-1)+...nterms$$$$=(10+100+1000+...nterms)-n$$$$=\frac{10({10}^n-1)}{10-1}-n$$$$=\frac{10({10}^n-1)-9n}{9}$$$$S_n=\frac{{10}^{n+1}-9n-10}{81}$$

Answered by Ar Brandon last updated on 29/Aug/22

$$u_n=1+11+111+1111+...$$$$\Rightarrow u_n=\frac{{10}^n-1}{9}$$$$\Rightarrow \mathrm{\Sigma }{u}_n=\frac{10}{9}\left(\frac{{10}^n-1}{9}\right)-\frac{n}{9}$$

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