Question

[Vo Thi Minh Dao]

rút gọn biểu thức A=75\(\left(4^{1993}+4^{1992}+...+4^2+5\right)\) +25

Answer 1

[Trang Thùy]

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

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

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

Mà: \(A=75B+25=25\left(3B+1\right)=25\left(4^{1994}-1+1\right)=25.4^{1994}\)

Answer 2

[Nguyễn Việt Lâm]

\(A=75\left(4^{1993}+5^{1992}+...+4^2+5\right)+25=75B+25\)

Xét \(B=4^{1993}+4^{1992}+...+4^2+5=4^{1993}+4^{1992}+...+4^2+4+1\)

\(\Rightarrow4B=4^{1994}+4^{1993}+...+4^2+4\)

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

\(\Rightarrow3B=4^{1994}-1\Rightarrow B=\dfrac{4^{1994}-1}{3}\)

Vậy \(A=75.\dfrac{\left(4^{1994}-1\right)}{3}+25=25.4^{1994}-25+25\)

\(\Rightarrow A=25.4^{1994}\)