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Manipal MET2017MathematicsSequences and Series

0.2+0.22+0.222+ . to n terms is equal to

Options

  1. A( 2 9 )- ( 2 81 ) (1-10^ -n )
  2. Bn- ( 1 9 ) (1-10^ -n )
  3. C( 2 9 ) [n- ( 1 9 ) (1-10^ -n ) ]
  4. D( 2 9 )

Correct answer

C. ( 2 9 ) [n- ( 1 9 ) (1-10^ -n ) ]

Step-by-step solution

0.2+0.22+0.222+ . n terms =2(011+011+0.111+ n terms ) aligned & =2 ( 1 10 + 11 100 + 111 1000 + n terms ) & = 2 9 ( 9 10 + 99 100 + 999 1000 + n terms ) & = 2 9 (1- 1 10 +1- 1 100 +1- 1 1000 + n terms ) & = 2 9 [n- ( 1 10 + 1 100 + 1 1000 + n ) ] & = 2 9 [n- 1 10 1- ( 1 10 )^n . (1- 1 10 ) ] ] & = 2 9 [n- 1 10 10 9 ( 10^n-1 10^n ) ] & = 2 9 [n- 1 9 (1-10^ -n ) ] aligned

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