Question

[mù tạt]

tính 

\(A=\frac{-1}{3}+\frac{1}{3^2}-\frac{1}{3^3}+......+\frac{1}{3^{100}}-\frac{1}{3^{101}}\)

giúp tớ với tớ phải nộp ngay

Answer 1

[Đặng Viết Thái]

ta có :

\(3A=-1+\frac{1}{3}-\frac{1}{3^2}+...+\frac{1}{3^{99}}-\frac{1}{3^{100}}\)

\(\Rightarrow4A=-1-\frac{1}{3^{101}}\)

\(\Rightarrow4A=\frac{-3^{101}-1}{3^{101}}\)

\(\Rightarrow A=\left(\frac{-3^{101}-1}{3^{101}}\right):4\)

Answer 2

[Cúc Ba Tư cô nương]

\(A=\frac{-1}{3}+\frac{1}{3^2}-\frac{1}{3^3}+...+\frac{1}{3^{100}}-\frac{1}{3^{101}}\)

\(\Rightarrow3A=-1+\frac{1}{3}-\frac{1}{3^2}+...+\frac{1}{3^{99}}-\frac{1}{3^{100}}\)

\(\Rightarrow3A+A=4A\)

 \(=\left(-1+\frac{1}{3}-...-\frac{1}{3^{100}}\right)+\left(\frac{-1}{3}+...+\frac{1}{3^{100}}-\frac{1}{3^{101}}\right)\)

\(=-1+\frac{1}{3}-...-\frac{1}{3^{100}}-\frac{1}{3}+...+\frac{1}{3^{100}}-\frac{1}{3^{101}}\)

\(=-1-\frac{1}{3^{101}}\)

\(\Rightarrow A=\frac{-1-\frac{1}{3^{101}}}{4}\)

Vậy \(A=\frac{-1-\frac{1}{3^{101}}}{4}\)