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If the sum 3 1 2 + 5 1 2 + 2 2 + 7 1 2 + 2 2 + 3 2 + .... . + up to 20 terms is equal to k 21 , then k is equal to

Options

  1. A240
  2. B120
  3. C60
  4. D180

Correct answer

B. 120

Step-by-step solution

S 20 = 3 1 2 + 5 1 2 + 2 2 + 7 1 2 + 2 2 + 3 2 + . ... + upto   20 terms = Σ r = 1 20 2 r + 1 1 2 + 2 2 + 3 2 + . ...  r 2 = Σ r = 1 20 2 r + 1 r 6 r + 1 2 r + 1 = 6 Σ r = 1 20 1 r r + 1 = 6 Σ r = 1 20 1 r - 1 r + 1 = 6 1 1 - 1 2 + 1 2 - 1 3 + 1 3 - 1 4 + . ... + 1 20 - 1 21 = 6 1 - 1 21 = 6 × 20 21 = 120 21 = k 21 k = 120

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