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Manipal MET2013MathematicsSequences and Series

The sum to n terms of the series: 1 1+1^2+1^4 + 2 1+2^2+2^4 + 3 1+3^2+3^4 + is

Options

  1. An^2+1 2 (n^2+n+1 )
  2. Bn^2+n (n^2+n+1 )
  3. Cn^2+n 2 (n^2+n+1 )
  4. DNone of these

Correct answer

C. n^2+n 2 (n^2+n+1 )

Step-by-step solution

Let T_f be the r^ th term of the given series. Then, T_r= r 1+r^2+r^4 , r=1,2,3, , n aligned & = r (r^2+r+1 ) (r^2-r+1 ) & = 1 2 [ 1 r^2-r+1 - 1 r^2+r+1 ] aligned Sum of the series = _ r=1 ^n T_r= 1 2 _ r=1 ^n ( 1 r^2-r+1 - 1 r^2+r+1 ) array r = 1 2 (1- 1 3 )+ ( 1 3 - 1 7 )+ ( 1 7 - 1 13 )+ + . . ( 1 n^2-n+1 - 1 n^2+n+1 ) array = 1 2 1- 1 n^2+n+1 = n^2+n 2 (n^2+n+1 )

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