Question

[Vương Quốc Anh]

Rút gọn biểu thức: 

A=75(4¹⁹⁹³+4¹⁹⁹²+…+4²+5)+25

 

Answer 1

[Lê Nguyên Hạo]

Đặt B = 4^1993 + 4^1992 + ... + 4^2 + 4 + 1 
=> 4B = 4^1994 + 4^1993 + ... + 4^3 + 4^2 + 4 
=> 3B = 4^1994 - 1 
Ta lại có: 
A = 75B + 25 = 25.(3B+1) = 25.(4^1994 - 1 + 1) = 25.4^1994

Answer 2

[Nguyễn Huy Tú]

Đặt \(B=4^{1993}+4^{1992}+...+4^2+5\)

\(B=4^{1993}+4^{1992}+...+4^2+4+1\)

\(\Rightarrow B=1+4^2+...+4^{1992}+4^{1993}\)

\(\Rightarrow4B=4+4^3+...+4^{1993}+4^{1994}\)

\(\Rightarrow4B-B=\left(4+4^3+...+4^{1993}+4^{1994}\right)-\left(1+4^2+...+4^{1992}+4^{1993}\right)\)

\(\Rightarrow3B=\left(4^{1994}+4-1\right)\)

\(\Rightarrow3B=4^{1994}+3\)

\(\Rightarrow B=\left(4^{1994}+3\right):3\)

\(\Rightarrow A=75.\left(4^{1994}+3\right):3+25\)

\(\Rightarrow A=25.\left(4^{1994}+3\right)+25\)

\(\Rightarrow A=25.\left(4^{1994}+3+1\right)\)

\(\Rightarrow A=25.\left(4^{1994}+4\right)\)

\(\Rightarrow A=4^{1994}.25+100\)

Vậy \(A=4^{1994}.25+100\)