1. Tardigrade
  2. Question
  3. Mathematics
  4. The sum of 10 terms of the series 0.7+.77+.777+ ldots ldots ldots is-

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The sum of 10 terms of the series $0.7+.77+.777+\ldots \ldots \ldots$ is-

Sequences and Series

Solution:

The sum of 10 terms of series $.7+.77+.777+\ldots \ldots$
$S_{10}=+\frac{7}{10}+\frac{77}{100} \ldots \ldots \ldots$. to 10 terms
$=\frac{7}{9}\left[\frac{9}{10}+\frac{99}{100}+\ldots \ldots \ldots\right.$.to 10 terms $]$
$=\frac{7}{9}\left[\frac{10-1}{10}+\frac{100-1}{100}+\frac{1000-1}{1000}+\ldots \ldots \ldots\right.$ to 10 terms $]$
$=\frac{7}{9}\left[(1+1+1 \ldots \ldots . .10\right.$ terms $)-\left(\frac{1}{10}+\frac{1}{10^2}+\frac{1}{10^3}+\ldots \ldots \ldots .\right.$. to 10 terms $\left.)\right]$
$=\frac{7}{9}\left[10-\frac{\frac{1}{10}\left(1-\left(\frac{1}{10}\right)^{10}\right)}{1-\frac{1}{10}}\right]=\frac{7}{9}\left[10-\frac{\left(10^{10}-1\right)}{9.10^{10}}\right]=\frac{7}{81}\left[90-1+\frac{1}{10^{10}}\right]=\frac{7}{81}\left[89+\frac{1}{10^{10}}\right]$