Toán tìm 2 chữ số tận cùng của 2+2^2+…..+2^99 17/09/2021 By Hailey tìm 2 chữ số tận cùng của 2+2^2+…..+2^99
\(\begin{array}{l} A = 2 + {2^2} + ……. + {2^{99}}\\ \Rightarrow 2A = 2\left( {2 + {2^2} + …… + {2^{99}}} \right) = {2^2} + {2^3} + …. + {2^{100}}\\ \Rightarrow 2A – A = {2^2} + {2^3} + …. + {2^{100}} – \left( {2 + {2^2} + ……. + {2^{99}}} \right)\\ \Leftrightarrow A = {2^{100}} – 2\\ Ta\,\,co:\,\,\,{\left( {…2} \right)^{20}} = \overline {…76} \\ \Rightarrow {2^{100}} = {\left[ {{{\left( {…2} \right)}^{20}}} \right]^5} = {\left( {\overline {…76} } \right)^5} = \overline {…76} \\ \Rightarrow {2^{100}} – 2 = \overline {…76} – 2 = \overline {…74} \\ \Rightarrow A\,\,\,co\,\,\,chu\,\,\,\,so\,\,\,\tan \,\,cung\,\,\,la\,\,\,74. \end{array}\) Trả lời
\(\begin{array}{l}
A = 2 + {2^2} + ……. + {2^{99}}\\
\Rightarrow 2A = 2\left( {2 + {2^2} + …… + {2^{99}}} \right) = {2^2} + {2^3} + …. + {2^{100}}\\
\Rightarrow 2A – A = {2^2} + {2^3} + …. + {2^{100}} – \left( {2 + {2^2} + ……. + {2^{99}}} \right)\\
\Leftrightarrow A = {2^{100}} – 2\\
Ta\,\,co:\,\,\,{\left( {…2} \right)^{20}} = \overline {…76} \\
\Rightarrow {2^{100}} = {\left[ {{{\left( {…2} \right)}^{20}}} \right]^5} = {\left( {\overline {…76} } \right)^5} = \overline {…76} \\
\Rightarrow {2^{100}} – 2 = \overline {…76} – 2 = \overline {…74} \\
\Rightarrow A\,\,\,co\,\,\,chu\,\,\,\,so\,\,\,\tan \,\,cung\,\,\,la\,\,\,74.
\end{array}\)