Tính tổng \(S = 1 + 2.2 + {3.2^2} + {4.2^3} + ........ + {2018.2^{2017}}\)

* Hướng dẫn giải

Ta có \(A = 1 + 2 + {2^2} + {2^3} + ... + {2^n} = {2^{n + 1}} - 1\)

Xét \(2S = 1.2 + {2.2^2} + {3.2^3} + {4.2^4} + ... + {2017.2^{2017}} + {2018.2^{2018}}\)

Và \(S = 1 + 2.2 + {3.2^2} + {4.2^3} + ... + {2017.2^{2016}} + {2018.2^{2017}}\)

Suy ra

\(S = {2018.2^{2018}} - \left( {1 + 2 + {2^2} + {2^3} + ... + {2^{2017}}} \right)\)

\( = {2018.2^{2018}} - \left( {{2^{2018}} - 1} \right) = {2017.2^{2018}} + 1\)