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Manipal MET2013MathematicsSequences and Series

Sum of the series to n terms 5+7+13+31+85+ is

Options

  1. A3^n+8 n+1
  2. B1 2 [3^n+8 n-1 ]
  3. C1 2 (3^n+8 n+1 )
  4. DNone of the above

Correct answer

B. 1 2 [3^n+8 n-1 ]

Step-by-step solution

The sequence of differences between successive terms is 2,6,18,54, . Clearly, it is a GP. Let T_n be the n^ th term of the given series and S_n be the sum of its n terms. Then, S_n=5+7+13+31+85+ +T_ n-1 +T_n ....(i) S_n=5+7+13+31+ +I_ n-1 +I_n ...(ii) Subtracting (ii) from (i), we get aligned & O=5+ [2+6+18+54+ + (T_n-T_ n-1 ) ]-T_n & O=5+2 3^ n-1 -1 3-1 -T_n & T_n=5+ (3^ n-1 -1 )=4+3^ n-1 & aligned & S_n= _ k=1 ^n T_k= _ k=1 ^n (4+3^ k-1 ) &= _ k=1 ^n 4+ _ k=1 ^n 3^ k-1 & =4 n+ (1+3+3^2+ +3^ n-1 ) aligned aligned

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