Question

[Nguyễn Hà Khắc]

Rút gọn biểu thức trên :

A =\(2\times2^2+3\times2^3+4\times2^4+5\times2^5+...+100\times2^{100}\)

Answer 1

[Hisu Hydrangea]

A=\(2.2^2+3.2^3+4.2^4+...+100.2^{100}\)

\(\Rightarrow2A=2.2^3+3.2^4+4.2^5+...+100.2^{101}\)

\(\Rightarrow A-2A=2.2^2+\left(3.2^3-2.2^3\right)+\left(4.2^4-3.2^4\right)+...+\left(100.2^{100}-99.2^{100}\right)-100.2^{101}\)

\(\Rightarrow-A=2^3+\left(2^3+2^4+...+2^{100}\right)-100.2^{101}\)

Đặt \(B=\left(2^3+2^4+...+2^{100}\right)\)

\(\Rightarrow2B=\left(2^4+2^5+...+2^{101}\right)\)

\(\Rightarrow2B-B=\left(2^4+2^5+...+2^{101}\right)-\left(2^3+2^4+...+2^{100}\right)\)

\(\Rightarrow B=2^{101}-2^3\)

\(\Rightarrow-A=2^3+2^{101}-2^3-100.2^{101}\)

\(\Rightarrow-A=2^{101}-100.2^{101}\)

\(\Rightarrow A=100.2^{101}-2^{101}=99.2^{101}\)