Tính \(B = \frac{1}{2} + {\left( {\frac{1}{2}} \right)^2} + {\left( {\frac{1}{2}} \right)^3} + {\left( {\frac{1}{2}} \right)^4} + ..... + {\left( {\frac{1}{2}} \right)^{98}} + {\left( {\frac{1}{2}} \right)^{99}}\)

Lời giải:

\(2B = 1 + \frac{1}{2} + {\left( {\frac{1}{2}} \right)^2} + {\left( {\frac{1}{2}} \right)^3} + {\left( {\frac{1}{2}} \right)^4} + ..... + {\left( {\frac{1}{2}} \right)^{97}} + {\left( {\frac{1}{2}} \right)^{98}}\)

\(\begin{array}{l}2B - B = \left[ {1 + \frac{1}{2} + {{\left( {\frac{1}{2}} \right)}^2} + {{\left( {\frac{1}{2}} \right)}^3} + {{\left( {\frac{1}{2}} \right)}^4} + ..... + {{\left( {\frac{1}{2}} \right)}^{97}} + {{\left( {\frac{1}{2}} \right)}^{98}}} \right]\\ - \left[ {\frac{1}{2} + {{\left( {\frac{1}{2}} \right)}^2} + {{\left( {\frac{1}{2}} \right)}^3} + {{\left( {\frac{1}{2}} \right)}^4} + ..... + {{\left( {\frac{1}{2}} \right)}^{98}} + {{\left( {\frac{1}{2}} \right)}^{99}}} \right]\end{array}\)

\(B = 1 - {\left( {\frac{1}{2}} \right)^{90}}\)