If 1+(1-22 ⋅ 1)+(1-42 ⋅ 3)+(1-62 ⋅ 5)+ ldots ldots+(1-202 ⋅ 19) =α-220 β, then an ordered pair (α, β) is equal to :

Question

If 1+(1-22 ⋅ 1)+(1-42 ⋅ 3)+(1-62 ⋅ 5)+ ldots ldots+(1-202 ⋅ 19) =α-220 β, then an ordered pair (α, β) is equal to : (A) (10,97) (B) (11,103) (C) (10,103) (D) (11,97)

Tardigrade Admin · Accepted Answer

1+(1-22 ⋅ 1)+(1-42 ⋅ 3)+ ldots ldots+(1-202 ⋅ 19) =α-220 β =11-(22 ⋅ 1+42 ⋅ 3+ ldots ldots+202 ⋅ 19) =11-22 ⋅ displaystyle∑ r =110 r 2(2 r -1)=11-4((1102/2)-35 × 11) =11-220(103) ⇒ α=11, β=103

Q. If 1+(1−22⋅1)+(1−42⋅3)+(1−62⋅5)+……+(1−202⋅19) =α−220β, then an ordered pair (α,β) is equal to :

(10,97)

(11,103)

(10,103)

(11,97)

1+(1−22⋅1)+(1−42⋅3)+……+(1−202⋅19) =α−220β =11−(22⋅1+42⋅3+……+202⋅19) =11−22⋅r=1∑10​r2(2r−1)=11−4(21102​−35×11) =11−220(103) ⇒α=11,β=103

Q. If $1 + (1 - 2^{2} \cdot 1) + (1 - 4^{2} \cdot 3) + (1 - 6^{2} \cdot 5) + \dots\dots + (1 - 2 0^{2} \cdot 19)$ $= α - 220 β,$ then an ordered pair $(α, β)$ is equal to :