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Sum to n terms of the series 1 4 1.3 + 2 4 3.5 + 3 4 5.7 + . . . . . is equal to

Options

  1. An n + 1 2 n 2 + 1 8 2 n + 1
  2. Bn n + 1 n 2 + n + 1 6 2 n + 1
  3. Cn + 1 2 n + 1 2 + 1 8 2 n + 1
  4. Dn n + 1 2 n + 1 2 + 1 16 2 n + 1

Correct answer

B. n n + 1 n 2 + n + 1 6 2 n + 1

Step-by-step solution

The k t h term of the given series is u k = k 4 2 k - 1 2 k + 1 = k 2 4 + 1 16 + 1 32 1 2 k - 1 - 1 2 k + 1 Therefore the sum of n terms is S n = u 1 + u 2 + . . . . + u n = 1 4 ∑ k = 1 n k 2 + n 16 + 1 32 1 - 1 2 n + 1 = n n + 1 2 n + 1 4.6 + n 16 + n 16 2 n + 1 = n n + 1 n 2 + n + 1 6 2 n + 1

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