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What is the sum of the series 0.5+0.55+0.555+ to n terms?

Options

  1. A5 9 [n- 2 9 (1- 1 10^n ) ]
  2. B1 9 [5- 2 9 (1- 1 10^n ) ]
  3. C1 9 [n- 5 9 (1- 1 10^n ) ]
  4. D5 9 [n- 1 9 (1- 1 10^n ) ]

Correct answer

D. 5 9 [n- 1 9 (1- 1 10^n ) ]

Step-by-step solution

Given 0.5 + 0.55 + 0.555 + ..... to n = 5 [0.1 + 0.11 + 0.111 + ..... to n terms] aligned & = 5 9 [0.9+0.99+0.999+ . . to n terms ] & = 5 9 [ 9 10 + 99 100 + 999 1000 + . to n terms ] aligned aligned & = 5 9 ( (1- 1 10 )+ (1- 1 10^2 )+ (1- 1 10^3 )+ . . & . (1- 1 10^ n ) ] & = 5 9 [ n - ( 1 10 + 1 10^2 + 1 10^ n ) ] aligned = 5 9 [ n - 1 10 1- ( 1 10 )^ n (1- 1 10 ) ]= 5 9 [ n - 1 9 (1- 1 10^ n ) ]

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