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KCET2013MathematicsSequences and Series

The sum of the series, 1 2.3 2+ 2 3.4 2²+ 3 4.5 2³+ upto n terms is

Options

  1. A2^ n+1 n+2 +1
  2. B2^ n+1 n+2 -1
  3. C2^ n+1 n+2 +2
  4. D2^ n+1 n+2 -2

Correct answer

B. 2^ n+1 n+2 -1

Step-by-step solution

Given series is 1 2 3 2¹+ 2 3.4 2²+ 3 4.5 2³+ upto n terms The n th term of the series is T_ n = n (n+1)(n+2) 2^ n = 2 2^ n (n+2) - 2^ n (n+1) On putting n=1,2,3 , we get T₁=2 2¹ 3 - 2¹ 2 , T₂= 2 2² 4 - 2² 3 , T₃= 2 2³ 5 - 2³ 4 On adding, we get aligned & S_ n =T₁+T₂+T₃+ +T_ n & S_ n = 2 2^ n (n+2) - 2 2 = 2^ n+1 (n+2) -1 aligned

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