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If _ r=1 ^n T_r= (2 n-1)(2 n+1)(2 n+3)(2 n+5) 64 , then _ n _ r=1 ^n ( 1 T_r ) is equal to :

Options

  1. A0
  2. B2 3
  3. C1
  4. D1 3

Correct answer

B. 2 3

Step-by-step solution

aligned & T _ n = S _ n - S _ n -1 & ~T _ n = 1 8 (2 n -1)(2 n +1)(2 n +3) & 1 ~T _ n = 8 (2 n -1)(2 n +1)(2 n +3) & _ n _ r =1 ^ n 1 ~T _ r = _ n 8 _ r =1 ^ n 1 (2 n -1)(2 n +1)(2 n +3) & = _ n 8 4 ( 1 (2 n -1)(2 n +1) - 1 (2 n +1)(2 n +3) ) & = _ n 2 [ ( 1 1.3 - 1 3.5 )+ ( 1 3.5 - 1 5.7 )+ ] & = 2 3 aligned

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