If 1 + 1 - 2 2 · 1 + 1 - 4 2 · 3 + 1 - 6 2 · 5 + … … + 1 - 20 2 · 19 = &#945…

If 1+1-22·1+1-42·3+1-62·5+……+1-202·19=α-220 β, then an ordered pair α, β is equal to:
  1. 10, 97
  2. (11, 103)
  3. 10, 103
  4. 11, 97

Solution

S=1+∑r=1101−2r22r−1

=1+10−∑r=1108r3−4r2

=11−810×1122−410×11×216

=11−21102−140×11

=11−221100−70

=11−220110−7

∴ α−220β=11−220103

∴ α=11, β=103

α, β=11, 103.

Asked in: JEE Main 2020 (04 Sep Shift 1)

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