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Question Number 172687 by Rasheed.Sindhi last updated on 30/Jun/22

$$\mathrm{Sum}\mathrm{to}\mathrm{n}\mathrm{terms}:$$$$7+77+777+...$$

Commented by mr W last updated on 30/Jun/22

$$S=1+11+111+...+\underset{ntimes}{111\cdot \cdot \cdot 1}$$$$10S=10+110+1110+...+\underset{ntimes}{111\cdot \cdot \cdot 10}$$$$-9S=n-10\times \underset{ntimes}{111\cdot \cdot \cdot 1}$$$$S=(10\times \underset{ntimes}{111\cdot \cdot \cdot 1}-n)/9$$$$$$$$7+77+777+...$$$$=7(1+11+111+...)$$$$=7S$$$$=\frac{7}{9}(10\times \underset{ntimes}{111\cdot \cdot \cdot 1}-n)$$

Commented by Rasheed.Sindhi last updated on 30/Jun/22

$$\mathcal{T}han\mathcal{X}sir!$$

Commented by Tawa11 last updated on 01/Jul/22

$$\mathrm{Great}\mathrm{sir}$$

Answered by som(math1967) last updated on 30/Jun/22

$$7(1+11+111+...)$$$$\frac{7}{9}(9+99+999+...)$$$$\frac{7}{9}(10-1+{10}^2-1+{10}^3-1+...)$$$$\frac{7}{9}\{(10+{10}^2+{10}^3+...)-1\times n\}$$$$\frac{7}{9}\times \frac{10({10}^n-1)}{10-1}-\frac{7n}{9}$$$$\frac{70({10}^n-1)}{81}-\frac{7n}{9}$$

Commented by Rasheed.Sindhi last updated on 30/Jun/22

$$\mathbb{T}\mathrm{han}\mathbb{k}\mathrm{s}\mathrm{sir}!$$

Commented by Tawa11 last updated on 01/Jul/22

$$\mathrm{Great}\mathrm{sir}$$

Commented by peter frank last updated on 03/Jul/22

$$\mathrm{thank}\mathrm{you}$$

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