Question

Tìm số nguyên n ,biết:

a,\(5^{-1}\times25^n=125\)

b,\(3^{-1}\times3^n+6\times3^{n-1}=7\times3^6\)

c,\(3^4<\frac{1}{9}\times27^n<3^{10}\)

d,\(25<5^n:5<625\)

Answer 1

25<5^n:5<625

=>5^2<5^n-1<5^4

=>2<n-1<4

=>n-1=3

=>n=4

Answer 2

a. \(\Rightarrow5^{-1}.5^{2n}=5^3\)

\(\Rightarrow5^{2n-1}=5^3\)

=> 2n-1=3

=> 2n=4

=> n=2

b. \(\Rightarrow3^{n-1}+6.3^{n-1}=7.3^6\)

\(\Rightarrow\left(1+6\right).3^{n-1}=7.3^6\)

\(\Rightarrow7.3^{n-1}=7.3^6\)

=> n-1=6

=> n=7

c. \(\Rightarrow3^4<3^{-2}.3^{3n}<3^{10}\)

\(\Rightarrow3^4<3^{3n-2}<3^{10}\)

\(\Rightarrow3n-2\in\left\{5;6;7;8;9\right\}\)

\(\Rightarrow3n\in\left\{7;8;9;10;11\right\}\)

\(\text{Mà n là số nguyên}\Rightarrow n=3\).

d. \(\Rightarrow5^2<5^{n-1}<5^4\)

\(\Rightarrow n-1=3\)

\(\Rightarrow n=4\).