The sum of first 20 terms of the sequence 0.7, 0.77, 0.777, .........., is :

Engineering

Mathematics

Geometric Progression and Its Properties

Question

$\frac{7}{9}$ (99 +10^–20)

$\frac{7}{9}$ (99 – 10^–20)

$\frac{7}{81}$ (179 – 10^–20)

$\frac{7}{81}$ (179 + 10^–20)

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S = 0.7 + 0.77 + 0.777 + .............20 terms

$= \frac{7}{10} + \frac{77}{100} + \frac{777}{1000} + . .......20 terms$

$= \frac{7}{9} [\frac{9}{10} + \frac{99}{10^{2}} + \frac{999}{10^{3}} + . .......20 terms]$

$S = \frac{7}{9} [(1 - \frac{1}{10}) + (1 - \frac{1}{100}) + (1 - \frac{1}{10^{3}}) + . ....20 terms]$

$= \frac{7}{9} [20 - (\frac{1}{10} + \frac{1}{10^{2}} + \frac{1}{10^{3}} + ........ \frac{1}{10^{20}})]$

$= \frac{7}{9} [20 - \frac{\frac{1}{10} (1 - {(\frac{1}{10})}^{20})}{1 - \frac{1}{10}}]$

$= \frac{7}{81} [179 + 10^{- 20}]$