# 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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