Question

Cho A=1/2²+1/2³+...+1/2¹⁰⁰

CMR: A+1/100=1/2

Answer 1

\(2A=2\left(\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{100}}\right)\)

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

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

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

Đến đây tôi chịu

Answer 2

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

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

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

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

Vậy \(A+\frac{1}{2^{100}}=\frac{1}{2}\)