Question

Answer this question without spamming :- • Find the sum of the first 20 terms of the sequence 0.7, 0.77, 0.777, ...

Answer 1

Answer:

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Answer 2

Given sequence :-

0.7, 0.77, 0.777, ...

To Find :-

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

Solution :-

We have -

◘ 0.7, 0.77, 0.777, ... upto 20 terms

→ 7(0.1 + 0.11 + 0.111 upto 20 terms)

$\displaystyle \sf{\longrightarrow 7\bigg(\frac{1}{10}+\frac{11}{100} +\frac{111}{1000}+...\ upto\ 20\ terms \bigg) }$

$\displaystyle \sf{\longrightarrow 7\bigg(\frac{1}{10}+\frac{11}{10^2} +\frac{111}{10^3}+...\ upto\ 20\ terms \bigg) }$

$\displaystyle \sf{\longrightarrow \frac{7}{9}\bigg(\frac{9}{10}+\frac{99}{10^2} +\frac{999}{10^3}+...\ upto\ 20\ terms \bigg) }$

$\displaystyle \sf{\longrightarrow \frac{7}{9}\bigg[\bigg(1-\frac{1}{10}\bigg)+\bigg( 1-\frac{1}{100} \bigg)+\bigg(1-\frac{1}{1000}\bigg)+...\ upto\ 20\ terms \bigg] }$

$\displaystyle \sf{\longrightarrow \frac{7}{9}\bigg[20-\bigg(\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+ ...\ upto\ 20\ terms \bigg) \bigg] }$

$\displaystyle \sf{\longrightarrow \frac{7}{9} \bigg\{20-\bigg[\frac{1}{10} \bigg(\frac{1- (\frac{1}{10})^{20}}{1-\frac{1}{10}} \bigg) \bigg] \bigg\} }$

$\displaystyle \sf{\longrightarrow \frac{7}{9}\bigg\{20-\frac{1}{9}\bigg[1- \bigg(\frac{1}{10} \bigg)^{20} \bigg] \bigg\} }$

$\displaystyle \sf{\longrightarrow \frac{7}{81}\bigg[179+\bigg(\frac{1}{10} \bigg) ^{20} \bigg] }\:\:\:\:\:\: \cdots\sf{\color{orange}{ANSWER}}$