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What is the sum of the series 0.3+0.33+0.333+ . n terms?

Options

  1. A1 3 [ n - 1 9 (1- 1 10^ n ) ]
  2. B1 3 [ n - 2 9 (1- 1 10^ n ) ]
  3. C1 3 [ n - 1 3 (1- 1 10^ n ) ]
  4. D1 3 [ n - 1 9 (1+ 1 10^ n ) ]

Correct answer

A. 1 3 [ n - 1 9 (1- 1 10^ n ) ]

Step-by-step solution

aligned & 0.3+0.33+0.333+ . n terms & = 3 10 + 33 100 + 333 1000 + & =3 ( 1 10 + 11 100 + 111 1000 + . ) & = 3 9 ( 9 10 + 99 100 + 999 1000 + . ) & = 1 3 [ (1- 1 10 )+ (1- 1 100 )+ (1- 1 1000 )+ . . ] & = 1 3 [ n - ( 1 10 + 1 10^2 + 1 10^3 + 1 10^ n ) ] & = 1 3 [n- 1 10 (1- 1 10^ n ) 1- 1 10 ] & ( Since, Sum of n terms of G.P = a (1-r^n ) 1-r ] & = 1 3 [n- 10^n-1 9 (10^n ) ]= 1 3 [n- 1 9 (1- 1 10^n ) ] aligned

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