Question

[Đỗ Mạnh Anh Hải]

Rút gọn biểu thức sau ạ:
75(4^2017+ 4^2016+ 4^2015+ ...+ 4^2+5)+25
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Answer 1

[Đan Anh]

Đặt \(A=75\left(4^{2017}+4^{2016}+4^{2015}+...+4^2+5\right)+25\)

\(B=4^{2017}+4^{2016}+4^{2015}+...+4^2+5\)

\(=4^{2017}+4^{2016}+4^{2015}+...+4^2+4+1\)

\(\Rightarrow4B=4^{2018}+4^{2017}+4^{2016}+...+4^3+4^2+4\)

\(\Rightarrow4B-B=\left(4^{2018}+4^{2017}+4^{2016}+...+4^3+4^2+4\right)-\left(4^{2017}+4^{2016}+4^{2015}+...+4^2+4+1\right)\)

\(3B=4^{2018}+4^{2017}+4^{2016}+...+4^3+4^3+4-4^{2017}-4^{2016}-4^{2015}-...-4^2-4-1\)

\(3B=4^{2018}-1\)

\(\Rightarrow B=\dfrac{4^{2018}-1}{3}\)

Suy ra: \(A=75.\dfrac{4^{2018}-1}{3}+25\)

\(A=25.\left(4^{2018}-1\right)+25\)

\(=25\left(4^{2018}-1+1\right)\)

\(=25.4^{2018}\)