NTA Abhyas JEE Main2020MathematicsSequences and SeriesPractice

The sum ∑ r = 1 50 r ⋅ 2 r + 2 50 - r is equal to

Options

A25 2 50 - 1
B50 2 50 - 1
C25 2 51 - 1
D50 2 51 - 1

Correct answer

D. 50 2 51 - 1

Step-by-step solution

∑ r = 1 50 r 2 r + 2 50 - r = 1 ⋅ 2 + 2 ⋅ 2 2 + 3 ⋅ 2 3 + . … … . . + 50 ⋅ 2 50 + 1 ⋅ 2 49 + 2 ⋅ 2 48 + 3 ⋅ 2 47 + . … . . + 50 ⋅ 2 0 = 2 0 0 + 50 + 2 1 1 + 49 + 2 2 2 + 48 + . … … . + 2 48 48 + 2 + 2 49 49 + 1 + 2 50 50 + 0 = 50 1 + 2 + 2 2 + . … . . + 2 49 + 2 50 = 50 2 51 - 1 2 - 1 = 50 2 51 - 1

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