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

\underset{―}{Trouver la somme de S_{n}}

Si S_{n} = 1 + 11 + 111 + \dots + 1111 \dots 111

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

S_{n} = 1 + 11 + 111 + \dots + 1111 \dots 11

9 S_{n} = 9 + 99 + 999 + \dots + 999 \dots 99

= (10 - 1) + (100 - 1) + (1000 - 1) + \dots n t e r m s

= (10 + 100 + 1000 + \dots n t e r m s) - n

= \frac{10 (10^{n} - 1)}{10 - 1} - n

= \frac{10 (10^{n} - 1) - 9 n}{9}

S_{n} = \frac{10^{n + 1} - 9 n - 10}{81}

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

u_{n} = 1 + 11 + 111 + 1111 + \dots

\Rightarrow u_{n} = \frac{10^{n} - 1}{9}

\Rightarrow Σ u_{n} = \frac{10}{9} (\frac{10^{n} - 1}{9}) - \frac{n}{9}