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If S n = ∑ r = 1 n t r = 1 6 n 2 n 2 + 9 n + 13 , then ∑ r = 1 n t r is equal to

Options

  1. A1 2 n n + 1
  2. B1 2 n n + 3
  3. Cn + 1 2
  4. Dn 2

Correct answer

B. 1 2 n n + 3

Step-by-step solution

t r = S r - S r - 1 = 1 6 r 2 r 2 + 9 r + 13 - 1 6 r - 1 2 r - 1 2 + 9 r - 1 + 13 = 2 6 r 3 - r - 1 3 + 9 6 r 2 - r - 1 2 + 13 6 r - r - 1 = 1 3 3 r 2 - 3 r + 1 + 3 2 2 r - 1 + 13 6 = r 2 + 2 r + 1 = r + 1 2 ⇒ t r = r + 1 ⇒ ∑ r = 1 n t r = ∑ r = 1 n r + 1 = ∑ r = 1 n r + ∑ r = 1 n 1 = n n + 1 2 + n = n n + 3 2

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