Ta có \(A=\frac{3}{5^3}+\frac{4}{5^4}+...+\frac{102}{5^{102}}+\frac{103}{5^{103}}\)

=> 5A = \(\frac{3}{5^2}+\frac{4}{5^3}+...+\frac{102}{5^{101}}+\frac{103}{5^{102}}\)

Khi đó 5A - A = \(\left(\frac{3}{5^2}+\frac{4}{5^3}+...+\frac{102}{5^{101}}+\frac{103}{5^{102}}\right)-\left(\frac{3}{5^3}+\frac{4}{5^4}+...+\frac{102}{5^{102}}+\frac{103}{5^{103}}\right)\)

=> 4A = \(\frac{3}{5^2}+\left(\frac{1}{5^3}+\frac{1}{5^4}+...+\frac{1}{5^{102}}\right)-\frac{103}{5^{103}}\)

=> 4A = \(\frac{3}{5^2}+\frac{\frac{1}{5^2}-\frac{1}{5^{102}}}{4}-\frac{103}{5^{103}}\)

=> A = \(\frac{3}{5^2.4}+\left(\frac{1}{5^2}-\frac{1}{5^{102}}\right).\frac{1}{16}-\frac{103}{5^{103}.4}\)

=> A = \(\frac{3}{100}+\frac{1}{5^2}.\frac{1}{16}\left(1-\frac{1}{5^{100}}\right)-\frac{103}{5^{103}.4}=\frac{3}{100}+\frac{1}{400}\left(1-\frac{1}{5^{100}}\right)-\frac{103}{5^{103}.4}\)

       \(=\frac{3}{100}+\frac{1}{400}-\frac{1}{400.5^{100}}-\frac{103}{5^{103}.4}=\frac{13}{400}-\frac{1}{400.5^{100}}-\frac{103}{5^{103}.4}< \frac{13}{400}\left(\text{ĐPCM}\right)\)

Vậy \(A< \frac{13}{400}\)(đpcm)

a) Diện tích tam giác ABC (Heron)

\(S_{ABC}=\frac{1}{4}\sqrt{\left(AB+BC+AC\right)\left(AB+BC-AC\right)\left(BC+AC-AB\right)\left(AC+AB-BC\right)}\)

\(S_{ABC}=\frac{1}{4}\sqrt{\left(6+10+8\right)\left(6+10-8\right)\left(10+8-6\right)\left(8+6-10\right)}=24\left(cm^2\right)\)

b)Xét tam giác ABC có 

\(BC^2=10^2=100\left(cm\right)\)

\(AB^2+AC^2=6^2+8^2=100\left(cm\right)\)

Vì 100cm=100cm

\(\Rightarrow BC^2=AB^2+AC^2\)

=> Tam giác ABC vuông tại A 

Xét diện tích tam giác ABC thường \(S_{ABCt}=\frac{AH.BC}{2}\left(1\right)\)

Xét diện tích tam giác ABC vuông \(S_{ABCv}=\frac{AC.AB}{2}\left(2\right)\)

Từ (1) và (2) 

\(\Leftrightarrow AH.BC=AB.AC\)

\(\Leftrightarrow AH.10=8.6\Leftrightarrow AH=4,8\left(cm\right)\)

Xét tam giác ABH vuông tại H 

\(\Rightarrow BH^2=AB^2-AH^2\left(PYTAGO\right)\)

\(\Rightarrow BH=\sqrt{AB^2-AH^2}\)

\(\Rightarrow BH=\sqrt{6^2-13,3^2}=3,6\left(cm\right)\)

Xét tam giác ACH vuông tại H

\(\Rightarrow HC^2=AC^2-AH^2\left(PYTAGO\right)\)

\(\Rightarrow HC=\sqrt{AC^2-AH^2}\)

\(\Rightarrow HC=\sqrt{8^2-4,8^2}=6,4\left(cm\right)\)

bút chì đọc tiếng anh là gì ?

Từ \(\frac{2019a+2020c}{2019a-2021c}=\frac{2019b+2020d}{2019b-2021d}\)

<=> \(\frac{2019a-2021c+4041c}{2019a-2021c}=\frac{2019b-2021d+4041d}{2019b-2021d}\)

<=> \(1+\frac{4041c}{2019a-2021c}=1+\frac{4041d}{2019b-2021d}\)

<=> \(\frac{4041c}{2019a-2021c}=\frac{4041d}{2019b-2021d}\)

<=> 4041c( 2019b - 2021d ) = 4041d( 2019a - 2021c )

<=> c( 2019b - 2021d ) = d( 2019a - 2021c )

<=> 2019bc - 2021dc = 2019ad - 2021cd

<=> 2019bc - 2021dc - 2019ad + 2021cd = 0

<=> 2019( bc - ad ) = 0

<=> bc - ad = 0

<=> bc = ad

<=> a/b = c/d

Đặt \(\frac{a}{b}=\frac{c}{d}=k\Rightarrow\hept{\begin{cases}a=kb\\c=kd\end{cases}}\)

Ta có : \(\left(\frac{a+b}{c+d}\right)^{2020}=\left(\frac{kb+b}{kd+d}\right)^{2020}=\left[\frac{b\left(k+1\right)}{d\left(k+1\right)}\right]^{2020}=\left(\frac{b}{d}\right)^{2020}=\frac{b^{2020}}{d^{2020}}\)(1)

\(\)\(\frac{a^{2020}+b^{2020}}{c^{2020}+d^{2020}}=\frac{\left(kb\right)^{2020}+b^{2020}}{\left(kd\right)^{2020}+d^{2020}}=\frac{k^{2020}b^{2020}+b^{2020}}{k^{2020}d^{2020}+d^{2020}}=\frac{b^{2020}\left(k^{2020}+1\right)}{d^{2020}\left(k^{2020}+1\right)}=\frac{b^{2020}}{d^{2020}}\)(2)

Từ (1) và (2) ta có đpcm

\(\frac{13}{40}-\frac{11}{47}.\frac{1}{2}+\frac{1}{20}-\)\(\frac{121}{47.72}\)

\(=\left(\frac{13}{40}+\frac{1}{20}\right)-\)\(\left(\frac{11}{47}.\frac{1}{2}+\frac{121}{47.72}\right)\)

\(=\left(\frac{13}{40}+\frac{2}{40}\right)\)\(-\left(\frac{11}{47}.\frac{1}{2}+\frac{11}{47}.\frac{11}{72}\right)\)

\(=\frac{3}{8}-\frac{11}{47}.\left(\frac{1}{2}+\frac{11}{72}\right)\)

\(=\frac{3}{8}-\frac{11}{47}.\frac{47}{72}\)

\(=\frac{3}{8}-\frac{11}{72}\)

\(=\frac{27}{72}-\frac{11}{72}\)

\(=\frac{2}{9}\)

1.     A. headache      B. architect     C. chemical          D. children 

2.     A. although      B. enough       C. paragraph       D. cough

3.     A. heard            B. pearl            C. heart              D. earth

4.     A. put               B. junk             C. adult                D. sun

5.     A. future            B. circus          C. button              D. suggest

6.     A. church           B. amchair      C. architect         D. children

7.     A. banana          B. sofa            C. away               D. occasion

Ta có : \(\frac{x}{2}=\frac{y}{5}=\frac{z}{7}\)=))  \(x=2k;y=5k;z=7k\)

Thay vào biểu thức A ta được : 

\(A=\frac{2k-5k+7z}{2k+2.5k-7k}=\frac{4k}{5k}=\frac{4}{5}\)

\(\frac{x}{2}=\frac{y}{5}=\frac{z}{7}=\frac{x-y+z}{2-5+7}=\frac{x-y+z}{4}\) (1)

\(\frac{x}{2}=\frac{y}{5}=\frac{z}{7}=\frac{x}{2}=\frac{2y}{10}=\frac{z}{7}=\frac{x+2y-z}{2+10-7}=\frac{x+2y-z}{5}\) (2)

Từ (1) và (2) \(\Rightarrow\frac{x-y+z}{4}=\frac{x+2y-z}{5}\Rightarrow A=\frac{x-y+x}{x+2y-z}=\frac{4}{5}\)