Đáp án :
Với `x = 1/2`
`⇔ A (1/2) = 1/2 + (1/2)^2 + (1/2)^3 + ... + (1/2)^{99} + (1/2)^{100}`
`⇔ A(1/2) = 1/2 + 1/2^2 + 1/2^3 + ... + 1/2^{99} + 1/2^{100}`
`⇔ 1/2 A (1/2) = 1/2 (1/2 + 1/2^2 + 1/2^3+...+1/2^{99} + 1/2^{100})`
`⇔ 1/2 A (1/2) = 1/2^2 + 1/2^3 + 1/2^4 + ... + 1/2^{100} + 1/2^{101} (1)`
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Đặt `B = 1/2^2 + 1/2^3 + 1/2^4 + ... + 1/2^{100} + 1/2^{101}`
`⇔ 2 B = 1/2 + 1/2^2 + 1/2^3 + ... + 1/2^{99} + 1/2^{100}`
`⇔ 2B - B = (1/2 + 1/2^2 + 1/2^3 + ... + 1/2^{99} + 1/2^{100}) - (1/2^2 + 1/2^3 + 1/2^4 + ... + 1/2^{100} + 1/2^{101})`
`⇔ B = 1/2 - 1/2^{101} (2)`
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Thay `(2)` vào `(1)` ta được :
`1/2 A (1/2) = 1/2 - 1/2^{101}`
`⇔ A(1/2) = (1/2 - 1/2^{101} ) : 1/2`
`⇔ A (1/2) = 1 - 1/2^{100}`
Vậy `A (1/2) = 1 - 1/2^{100}`
