JEE Main20198 Apr 2019Evening ShiftMathematicsSequences and SeriesActual
The sum ∑ k = 1 20 k 1 2 k is equal to
Options
- A1 – 11 2 20
- B2 – 21 2 20
- C2 – 3 2 17
- D2 – 11 2 19
Correct answer
D. 2 – 11 2 19
Step-by-step solution
S =   ∑ k = 1 20 k 2 k = 1 2 + 2 2 2 + 3 2 3 + 4 2 4 + … . . + 20 2 20 1 2   S =   ∑ k = 1 20 k 2 k + 1 = 1 2 2 + 2 2 3 + 3 2 4 + … + 19 2 20 + 20 2 2 By subtraction, 1 2   S = 1 2 + 1 2 2 + 1 2 3 +   … … . + 1 2 20 – 20 2 21 = 1 2 1 - 1 2 20 1 - 1 2 - 20 2 21 (Using sum of n terms of G.P.) = 1 – 1 2 20   – 20 2 21 =   1   – 11 2 20 ⇒ S = 2 - 11 2 19