Question

Tính:

\(1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{99}}+\dfrac{1}{2^{100}}\)

giúp mk nha

Answer 1

Đặt:

\(S=1+\dfrac{1}{2}+\dfrac{1}{2^2}+.....+\dfrac{1}{2^{99}}+\dfrac{1}{2^{100}}\)

\(\Rightarrow2S=2\left(1+\dfrac{1}{2}+\dfrac{1}{2^2}+.....+\dfrac{1}{2^{99}}+\dfrac{1}{2^{100}}\right)\)

\(\Rightarrow2S=2+1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{98}}+\dfrac{1}{2^{99}}\)

\(\Rightarrow2S-S=\left(2+1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{98}}+\dfrac{1}{2^{99}}\right)-\left(1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{99}}+\dfrac{1}{2^{100}}\right)\)\(\Rightarrow S=2-\dfrac{1}{2^{100}}\)

Answer 2

Đặt

\(A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{99}}+\dfrac{1}{2^{100}}\)

\(A.2=2+1+\dfrac{1}{2}+ ...+\dfrac{1}{2^{99}}+\dfrac{1}{2^{100}}\)

\(\Rightarrow A.2-A=2-\dfrac{1}{2^{100}}\)

\(\Rightarrow A=2-\dfrac{1}{2^{100}}\)

Hok tốt!