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

Solution :

0.7 + 0.77 + 0.777 + …… + upto 20 terms

= \(7\over 10\) + \(77\over 10^2\) + \(777\over 10^3\) +  ….. + upto 20 terms

= 7[ \(1\over 10\) +  \(11\over 10^2\) + \(111\over 10^3\) +  ….. + upto 20 terms ]

= \(7\over 9\)[ \(9\over 10\) +  \(99\over 100\) + \(999\over 1000\) +  ….. + upto 20 terms ]

= \(7\over 9\)[ (1 – \(1\over 10\)) +  (1 – \(1\over 10^2\)) + (1 – \(1\over 10^3\)) +  ….. + upto 20 terms ]

= \(7\over 9\){n – (\(1\over 10\) + \(1\over 10^2\) + ….. + \(1\over 10^n\)}

= \(7\over 9\)[n – \(1\over 10\)\((1 – ({1\over 10})^n)\over (1 – {1\over 10})\)]

= \(7\over 9\){\(9n –  1 + {1\over 10^n}\)}


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