A= 1/2 + 1/2² + 1/2³+ ... + 1/2¹⁰⁰

=>2A = 1+1/2+1/2² +...+ 1/2⁹⁹

=>2A -A= (1+1/2+1/2² +...+1/2⁹⁹) -(1/2+1/2²+1/2³ +..+1/2¹⁰⁰)

=> A = 1-1/2¹⁰⁰

a) \(9.27^n=3^5\Rightarrow3^2.\left(3^3\right)^n=3^5\)

\(\Rightarrow3^2.3^{3n}=3^5\Rightarrow3^{5n}=3^5\)

\(\Rightarrow5n=5\Rightarrow n=1\)

b)\(\left(2^3:4\right).2^n=4\Rightarrow\left(2^3:2^2\right).2^n=2^2\)

\(\Rightarrow2.2^n=2^2\Rightarrow2^{1+n}=2^2\)

\(\Rightarrow1+n=2\Rightarrow n=1\)

c)\(3^2.3^4.3^n=3^7\Rightarrow3^{6+n}=3^7\)

\(\Rightarrow6+n=7\Rightarrow n=1\)

d)\(2^{-1}.2^n+4.2^n=9.2^5\)

\(\Rightarrow2^n\left(2^{-1}+4\right)=3^2.2^5\)

\(\Rightarrow\)\(2^n\left(\frac{1}{2}+4\right)=3^2.2^5\)

\(\Rightarrow\)\(2^n.\frac{3^2}{2}=3^2.2^5\)

\(\Rightarrow\)\(2^{n-1}.3^2=3^2.2^5\)

\(\Rightarrow n-1=5\Rightarrow n=6\)

e)\(243\ge3^n\ge9.3^2\)

\(\Rightarrow3^5\ge3^n\ge3^2.3^2\)

\(\Rightarrow3^5\ge3^n\ge3^4\)

\(\Rightarrow5\ge n\ge4\Rightarrow5;4\)

f)\(2^{n+3}.2^n=128\)

\(\Rightarrow2^{n+3+n}=2^7\)

\(\Rightarrow2^{2n+3}=2^7\)

\(\Rightarrow2n+3=7\Rightarrow2n=4\Rightarrow n=2\)

Hok tối

/2,8-x/+1,2=/-4,6/

/2,8-x/       =4,6- 1,2

/2,8-x/       =3,4

=> 2,8-x=3,4 hoặc 2,8-x=-3,4

=> x=-0,6      hoặc  x=6,2

đề là sao bn

Ta có căn(x + 5) + 2/11 >= 2/11 (vì căn (x+5) >= 0)

Vậy A đạt giá trị nhỏ nhất là 2/11 khi và chỉ khi x = -5

 Ta có : 3/19 - 3.căn(x - 2) <= 3/19 ( vì -3.căn(x-2) <= 0)

Vậy B đạt giá  trị lớn nhất là 3/19 khi và chỉ khi x = 5

C = (căn - 3)/2 có giá trị nguyên nên (căn - 3) chia hết cho 2

Suy ra x là số chính phương lẻ

 Vì x < 50 nên x thuộc { 1^2;3^2;5^2;7^2} hay x thuộc {1;9;25;49}

\(\left(2x+1\right)^3=-64\)

\(\left(2x+1\right)^3=-4^3\)

\(2x+1=-4\)

\(2x=-4-1=-5\)

\(x=-\frac{5}{2}\)

\(\left(2x+1\right)^3=-64\)

\(=>\left(2x+1\right)^3=-4^3\)

\(=>2x+1=-4\)

\(=>2x=-4-1=-5\)

\(=>x=\frac{-5}{2}\)

Giả sử  \(x=a;y=b;z=c\)

Ta có : \(\frac{2x}{a}+\frac{3y}{b}+\frac{4z}{c}\ge9\sqrt[9]{\frac{x^2y^3z^4}{a^2b^3c^4}}\)

Mà \(\left(\frac{2x}{a}+\frac{3y}{b}+\frac{4z}{c}\right)^2\le\left(x^2+y^2+z^2\right)\left(\frac{4}{a^2}+\frac{9}{b^2}+\frac{16}{c^2}\right)\)

Xảy ra khi : \(\frac{ax}{2}=\frac{by}{3}=\frac{cz}{4}\Leftrightarrow\frac{a^2}{2}=\frac{b^2}{3}=\frac{c^2}{4}\)

Ta có hệ \(\hept{\begin{cases}\frac{a^2}{2}=\frac{b^2}{3}=\frac{c^2}{4}\\a^2+b^2+c^2\end{cases}\Leftrightarrow a=\frac{\sqrt{2}}{3};b=\frac{\sqrt{3}}{3};c=\frac{2}{3}}\)

Vậy \(P_{max}=\frac{32\sqrt{3}}{6561}\) khi \(x=\frac{\sqrt{2}}{3};y=\frac{\sqrt{3}}{3};z=\frac{2}{3}\)

Chúc bạn học tốt !!!

a)\(2x=3y,4y=5z\Leftrightarrow\frac{x}{3}=\frac{y}{2},\frac{y}{5}=\frac{z}{4}\Leftrightarrow\frac{x}{15}=\frac{y}{10},\frac{y}{10}=\frac{z}{8}\)

\(\Rightarrow\frac{x}{15}=\frac{y}{10}=\frac{z}{8}\Leftrightarrow\frac{2x}{30}=\frac{y}{10}=\frac{2z}{16}\)

ADTCDTS=NHAU TA CÓ

\(\frac{2x}{30}=\frac{y}{10}=\frac{2z}{16}=\frac{2x+y-2z}{30+10-16}=\frac{24}{24}=1\)

x=15

y=10

z=8

b) Ta có BCNN(2,3,4)=12

\(\Rightarrow\frac{2x}{12}=\frac{3x}{12}=\frac{4z}{12}\Leftrightarrow\frac{x}{6}=\frac{y}{4}=\frac{z}{3}\)

\(\Rightarrow\frac{x}{6}=\frac{y}{4}=\frac{z}{3}\Leftrightarrow\frac{x^2}{36}=\frac{y^2}{16}=\frac{z^2}{9}\)

ADTCDTS=NHAU TA CÓ

\(\frac{x^2}{36}=\frac{y^2}{16}=\frac{z^2}{9}=\frac{x^2+y^2+z^2}{36+16+9}=\frac{61}{61}=1\)

\(\frac{x^2}{36}=1\Rightarrow x^2=36\Rightarrow x=+_-6\)

\(\frac{y^2}{16}=1\Rightarrow x=+_-4\)

\(\frac{z^2}{9}=1\Rightarrow z=+_-3\)

TUỰ KẾT LUẬN NHA BẠN

C)\(\frac{x-6}{3}=\frac{y-8}{4}=\frac{z-10}{5}\Leftrightarrow\frac{x^2-36}{9}=\frac{y^2-64}{16}=\frac{z^2-100}{25}\)

ADTCDTS=NHAU TA CÓ

\(\frac{x^2-36}{9}=\frac{y^2-64}{16}=\frac{z^2-100}{25}=\frac{\left(x^2-36\right)+\left(y^2-64\right)+\left(z^2-100\right)}{9+16+25}\)

\(=\frac{x^2-36+y^2-64+z^2-100}{50}=\frac{\left(x^2+y^2+z^2\right)-\left(36-64-100\right)}{50}\)

\(=\frac{\left(x^2+y^2+z^2\right)-\left(36+64+100\right)}{50}=\frac{200-200}{50}=\frac{0}{50}=0\)

\(\Rightarrow\frac{x^2-36}{9}=0\Rightarrow x^2-36=0\Rightarrow x^2=36\Rightarrow x=+_-6\)

\(\frac{y^2-64}{16}=0\Rightarrow y^2-64=0\Rightarrow y^2=64\Rightarrow y==+_-8\)

\(\frac{z^2-100}{25}=0\Rightarrow z^2-100=0\Rightarrow z^2=100\Rightarrow z=+_-10\)

TỰ KẾT LUẠN NHA