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BITSAT2015MathematicsSequences and SeriesActual

The value of 3 4 + 15 16 + 63 64 + upto n terms is

Options

  1. An- 4^ n 3 - 1 3
  2. Bn + 4^ - n 3 - 1 3
  3. Cn+ 4^ n 3 - 1 3
  4. Dn - 4^ - n 3 + 1 3

Correct answer

B. n + 4^ - n 3 - 1 3

Step-by-step solution

Consider 3 4 + 15 16 + 63 64 + upto n terms = 2²-1 2² + 2⁴-1 2⁴ + 2⁶-1 2⁶ upto n term = (1- 1 2² )+ (1- 1 2⁴ )+ (1- 1 2⁶ )+ upton terms =(1+1+1+ . upto n terms )- ( 1 2² + 1 2⁴ + 1 2⁶ + . upto n terms )= n - 1 2² [ 1- ( 1 2² )^ n 1 1 2² ]= n - 1 3 (1-4^ - n )= n + 4^ - n 3 - 1 3

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