Ta có:\(S=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{9^2}\)

            \(=\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+...+\frac{1}{9.9}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{8.9}\)

Mà \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{8.9}\)

     \(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{8}-\frac{1}{9}\)

    \(=1-\frac{1}{9}\)   

      \(=\frac{8}{9}\)

Lại có \(S=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{9^2}\)

\(=\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+...+\frac{1}{9.9}>\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{9.10}\)

Mà        \(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{9.10}\)

\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{10}\)

\(=\frac{1}{2}-\frac{1}{10}\)

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

Vậy \(\frac{2}{5}< S< \frac{8}{9}\)

S< 1/1.2+1/2.3+1/3.4+...+1/8.9 = 1/1-1/2+1/2-1/3+1/3-1/4+...+1/8-1/9=1-1/9=8/9

=> S < 8/9

S> 1/2.3+1/3.4+1/4.5+...+1/9.10=1/2-1/3+1/3-1/4+1/4-1/5+...+1/9-1/10=1/2-1/10=4/10=2/5

=> S > 2/5

Đs: 2/5 < S < 8/9

de sai ak

@Nguyễn Huy Thắng Đề k sai, mình chứng minh được rồi -.-

Đặt \(A=1+2^2+2^3+...+2^8+2^9\)

\(=\left(1+2\right)+\left(2^2+2^3\right)+...+\left(2^8+2^9\right)\)

\(=3+2^2.\left(1+2\right)+...+2^8.\left(1+2\right)\)

\(=3+2^2.3+...+2^8.3\)

\(=3.\left(1+2^2+...+2^8\right)⋮3\)

\(\Rightarrow A⋮3\)

Bấm 0 khắc ra

*987435879876********-=-==-*9*-*==*87866544

VT = \(2\sqrt{2}\left(2-3\sqrt{3}\right)+\left(1-2\sqrt{2}\right)^2+6\sqrt{6}\)

\(=4\sqrt{2}-6\sqrt{6}+1-4\sqrt{2}+8+6\sqrt{6}=9\)=VP (đpcm)

S=(1+2)+(2^2+2^3)+(2^4+2^5)+(2^6+2^7)+(2^8+2^9)

  =1.(1+2)+2^2.(1+2)+2^4.(1+2)+2^6.(1+2)+2^8.(1+2)

  =1.3+2^2.3+2^4.3+2^6.3+2^8.3

  =3.(1+2^2+2^4+2^6+2^8) chia hết cho 3

S=1+2+2^2+2^3+2^4+2^5+2^6+2^7

S= (1+2) + (2^2+2^3) + (2^4+2^5) + (2^6+2^7)

S=3 + 3.4 + 3.16 + 3.64

S=255

Vì 255 chia hết cho 3

=> S sẽ chia hết cho 3

Người lạ ơi bố thí cho tôi ^_^

 \(S\) = 1 + 2 + 2²+ 2³ + 2⁴ + 2⁵ + 2⁶ + 2⁷ + 2⁸ + 2⁹

\(\Rightarrow\)\(S\)= 2⁰ + 2¹ + 2²+ 2³ + 2⁴ + 2⁵ + 2⁶ + 2⁷ + 2⁸ + 2⁹
\(\Rightarrow\)\(S\)= ( 2⁰ + 2¹ ) + ( 2²+ 2³) + ( 2⁴ + 2⁵ ) + ( 2⁶ + 2⁷ ) + ( 2⁸ + 2⁹ )

\(\Rightarrow\) \(S\)= 2⁰ . ( 2⁰ + 2¹ ) + 2² . ( 2⁰ + 2¹ ) + 2⁴ . ( 2⁰ + 2¹ ) + 2⁶ . ( 2⁰ + 2¹ ) + 2⁸ . ( 2⁰ + 2¹ )

\(\Rightarrow\)\(S\)= 2⁰ . 3 + 2² . 3 + 2⁴ . 3 + 2⁶ . 3 + 2⁸ . 3

\(\Rightarrow\)\(S\)= 3 . ( 2⁰ + 2² + 2⁴ + 2⁶ + 2⁸ ) \(⋮\)3 ( đpcm )