COMEDK2023Morning ShiftMathematicsSequences and SeriesActual
The sum of n terms of the series, 4 3 + 10 9 + 28 27 + is
Options
- A3^n(2 n+1)+1 2 (3^n )
- B3^n n-1 2 (3^n )
- C3^n(2 n+1)-1 2 (3^n )
- D3^n-1 2
Correct answer
C. 3^n(2 n+1)-1 2 (3^n )
Step-by-step solution
The given series is 4 3 + 10 9 + 28 27 + + a_n . The n -th term a_n can be written as 3^n + 1 3^n = 1 + 1 3^n = 1 + 3^ -n . The sum of n terms is S_n = _ k=1 ^ n a_k = _ k=1 ^ n (1 + 3^ -k ) = _ k=1 ^ n 1 + _ k=1 ^ n ( 1 3 )^k . The first part is _ k=1 ^ n 1 = n . The second part is a geometric progression with first term a = 1 3 and common ratio r = 1 3 . The sum of the geometric series is a(1 - r^n) 1 - r = 1 3 (1 - (1/3)^n) 1 - 1/3 = 1 3 (1 - 3^ -n ) 2/3 = 1 2 (1 - 3^ -n ) = 1 2 - 1 2 3^n . Thus, S_n = n + 1 2 -