Question

1) tính: 1+ 1/2+ 1/2^2+..........................+ 1/2^99 + 1/2^100 + 1/2^100

Answer 1

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

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

Ta có: \(2A=2\left(1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{99}}+\dfrac{1}{2^{100}}\right)\)

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

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

\(A=2-\dfrac{1}{2^{100}}\). Khi đó \(B=A+\dfrac{1}{2^{100}}=2-\dfrac{1}{2^{100}}+\dfrac{1}{2^{100}}=2\)