Question

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

$7 over 81 space left parenthesis 179 minus 10 to the power of negative 20 end exponent right parenthesis$
$7 over 9 left parenthesis 99 minus 10 to the power of negative 20 end exponent right parenthesis$
$7 over 81 left parenthesis 179 plus 10 to the power of negative 20 end exponent right parenthesis$
$7 over 9 left parenthesis 99 plus 10 to the power of negative 20 end exponent right parenthesis$

Solution

7 over 81 left parenthesis 179 plus 10 to the power of negative 20 end exponent right parenthesis

Let S = 0.7 + 0.77 +0.777 + .... upto 20 terms
$7 over 10 plus 77 over 10 squared plus 777 over 10 cubed plus..... space upto space 20 space terms space equals space open square brackets 1 over 10 space plus 11 over 10 squared plus 111 over 10 cubed plus........ plus space upto space 20 space terms close square brackets equals space 7 over 9 open square brackets 9 over 10 plus 99 over 100 plus 999 over 1000 plus..... plus space upto space 20 space terms close square brackets space equals 7 over 9 open square brackets open parentheses 1 minus 1 over 10 close parentheses plus open parentheses 1 minus 1 over 10 squared close parentheses plus open parentheses 1 minus 1 over 10 cubed close parentheses plus space..... upto space 20 space terms close square brackets$
$7 over 9 open square brackets left parenthesis 1 plus 1 plus..... upto space 20 space terms right parenthesis minus open parentheses 1 over 10 plus fraction numerator begin display style 1 end style over denominator begin display style 10 squared end style end fraction plus fraction numerator begin display style 1 end style over denominator begin display style 10 cubed end style end fraction plus.... upto space 20 space terms close parentheses close square brackets space equals space 7 over 9 space open square brackets 20 minus fraction numerator begin display style 1 over 10 open curly brackets 1 minus open parentheses 1 over 10 close parentheses to the power of 20 close curly brackets end style over denominator 1 minus begin display style 1 over 10 end style end fraction close square brackets space equals space 7 over 9 open square brackets 20 minus 1 over 9 open curly brackets 1 minus open parentheses 1 over 10 close parentheses to the power of 20 close curly brackets close square brackets space equals space 7 over 9 open square brackets 179 over 9 plus 1 over 9 open parentheses 1 over 10 close parentheses to the power of 20 close square brackets equals space 7 over 181 space open square brackets 179 plus left parenthesis 10 right parenthesis to the power of negative 20 end exponent close square brackets$

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