haizzz

\(m:3+m:9+m:27+m:81=\frac{5}{9}\)

\(m:\left(3+9+27+81\right)=\frac{5}{9}\)

\(m:120=\frac{5}{9}\)

\(m=\frac{5}{9}\times120\)

\(m=\frac{200}{3}\)

\(m.\frac{1}{3}+m.\frac{1}{9}+m.\frac{1}{27}+m.\frac{1}{81}=\frac{5}{9}\)

\(m\left(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}\right)=\frac{5}{9}\)

Vì: \(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}=\frac{27}{81}+\frac{9}{81}+\frac{3}{81}+\frac{1}{81}=\frac{40}{81}\)

nên \(m.\frac{40}{81}=\frac{5}{9}\)

\(m=\frac{5}{9}:\frac{40}{81}=\frac{5}{9}.\frac{81}{40}=\frac{9}{8}\)

\(\frac{3^{-m}}{81}=27\)

\(=>3^{-m}=27\cdot81\)

\(3^{-m}=2187\)

Vì nếu \(k^{-m}\) thì => k = \(\frac{1}{k^m}\)

mà 2187 \(\in N\)

=> Không tìm được m thỏa mãn yêu cầu đề bài.

\(\frac{3^{-m}}{81}=27\Rightarrow3^{-m}=27.81=2187=3^7\)

\(\Rightarrow-m=7\)

\(\Rightarrow m=-7\)

M = \(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...+\frac{1}{6561}\)

=> 3M = \(1+\frac{1}{3}+\frac{1}{9}+...+\frac{1}{2187}\)

=> 3M - M = ( \(1+\frac{1}{3}+\frac{1}{9}+...+\frac{1}{2187}\)  ) - ( \(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...+\frac{1}{6561}\))

2M = 1 - \(\frac{1}{6561}\)

2M = \(\frac{6560}{6561}\)

=> M = \(\frac{3280}{6561}\)

\(M=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+.......+\frac{1}{6561}\)

\(\Rightarrow M=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+.........+\frac{1}{3^8}\)

\(\Rightarrow3M=3\left(\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+.........+\frac{1}{3^8}\right)\)

\(\Rightarrow3M=1+\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+............+\frac{1}{3^7}\)

\(\Rightarrow3M-M=1+\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+..........+\frac{1}{3^7}-\frac{1}{3}-\frac{1}{3^2}-\frac{1}{3^3}-.......-\frac{1}{3^8}\)

\(\Rightarrow2M=1-\frac{1}{3^8}\)

\(\Rightarrow M=\frac{1-\frac{1}{3^8}}{2}\)

Vậy M = \(\frac{1-\frac{1}{3^8}}{2}\)

\(M=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+...+\frac{1}{6561}\)

\(\Rightarrow3M=3\left(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+...+\frac{1}{6561}\right)\)

\(\Rightarrow3M=1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...+\frac{1}{2187}\)

\(\Rightarrow3M-M=\left(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...+\frac{1}{2187}\right)-\left(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\frac{1}{6561}\right)\)

\(\Rightarrow2M=1-\frac{1}{6561}\)

\(\Rightarrow2M=\frac{6560}{6561}\)

\(\Rightarrow M=\frac{3280}{6561}\)

\(\frac{3^{-m}}{81}=27\)

\(\frac{3^{-m}}{3^4}=3^3\)

\(3^{-m}=3^3\times3^4\)

\(3^{-m}=3^7\)

\(-m=7\)

\(m=7\)

3^-m  = 3⁷

1/3^m = 3⁷

3^{m +7} = 1 = 3^o

m= -7

tài năng toán học đặng yến linh

3M=1+1/3+1/9+...+1/2187

2M=3M-M

2M=1-1/6561

2M=6560/6561

M=3280/6561

\(\frac{27^x}{3^x}=81\)

\(\Rightarrow\frac{\left(3^3\right)^x}{3^x}=81\)

\(\Rightarrow\frac{3^{3x}}{3^x}=81\)

\(\Rightarrow3^{3x-x}=81\)

\(\Rightarrow3^{2x}=81\)

\(\Rightarrow3^{2x}=3^4\)

\(\Rightarrow2x=4\)

\(\Rightarrow x=4:2\)

\(\Rightarrow x=2\)

Vậy \(x=2.\)

Chúc bạn học tốt!

Ta có :

81.27.3² = 3⁴.3³.3² = 3⁹

81³ : 27² = (3⁴)³ : (3³)² = 3¹² : 3⁶ = 3⁶

Mà 3⁹ > 3⁶ ⇒ 81. 27. 3² > 81³ : 27²