\(A=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+\frac{1}{16}+\frac{1}{18}+\frac{1}{20}\)

\(A=1,46\)

a) x⁴+x³+2x²+x+1=(x⁴+x³+x²)+(x²+x+1)=x²(x²+x+1)+(x²+x+1)=(x²+x+1)(x²+1)

b)a³+b³+c³-3abc=a³+3ab(a+b)+b³+c³ -(3ab(a+b)+3abc)=(a+b)³+c³-3ab(a+b+c)

=(a+b+c)((a+b)²-(a+b)c+c²)-3ab(a+b+c)=(a+b+c)(a²+2ab+b²-ac-ab+c²-3ab)=(a+b+c)(a²+b²+c²-ab-ac-bc)

c)Đặt x-y=a;y-z=b;z-x=c

a+b+c=x-y-z+z-x=o

đưa về như bài b

d)nhóm 2 hạng tử đầu lại và 2hangj tử sau lại để 2 hạng tử sau ở trong ngoặc sau đó áp dụng hằng đẳng thức dề tính sau đó dặt nhân tử chung

e)x²(y-z)+y²(z-x)+z²(x-y)=x²(y-z)-y²((y-z)+(x-y))+z²(x-y)

=x²(y-z)-y²(y-z)-y²(x-y)+z²(x-y)=(y-z)(x²-y²)-(x-y)(y²-z²)=(y-z)(x²-2y²+xy+xz+yz)

1)

a) \(x+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}=5\)

\(x+\frac{64}{128}+\frac{32}{128}+\frac{16}{128}+\frac{8}{128}+\frac{4}{128}+\frac{2}{128}+\frac{1}{128}=5\)

\(x+\frac{127}{128}=5\)

\(x=5-\frac{127}{128}=\frac{513}{128}\)

b) \(x+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\frac{1}{243}+\frac{1}{729}+\frac{1}{2187}=3\)

\(x+\frac{729}{2187}+\frac{243}{2187}+\frac{81}{2187}+\frac{27}{2187}+\frac{9}{2187}+\frac{3}{2187}+\frac{1}{2187}=3\)

\(x+\frac{2186}{2187}=3\)

\(x=3-\frac{2186}{2187}=\frac{4375}{2187}\)

2)

a) \(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+\frac{1}{5\cdot6}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}\)

\(=1-\frac{1}{6}=\frac{5}{6}\)

b) \(5\frac{1}{2}+3\frac{5}{6}+\frac{2}{3}\)

\(=\left(5+3\right)+\left(\frac{1}{2}+\frac{2}{3}+\frac{5}{6}\right)\)

\(=8+\left(\frac{3}{6}+\frac{4}{6}+\frac{5}{6}\right)\)

\(=8+2=10\)

c) \(7\frac{7}{8}+1\frac{4}{6}+3\frac{3}{5}\)

\(=\left(7+1+3\right)+\left(\frac{7}{8}+\frac{2}{3}+\frac{3}{5}\right)\)

\(=11+\left(\frac{105}{120}+\frac{80}{120}+\frac{72}{120}\right)\)

\(=11+\frac{257}{120}=\frac{1577}{120}\)

3) Gọi số đó là x. Theo đề ta có :

\(\frac{16-x}{21+x}=\frac{5}{7}\)

\(7\left(16-x\right)=5\left(21+x\right)\)

\(112-7x=105+5x\)

\(112-105=7x-5x\)

\(7=2x\)

\(x=\frac{7}{2}=3,5\) ( vô lí )

Vậy không có số tự nhiên để thõa mãn điều kiện trên.

\(A=\frac{1}{1.101}+\frac{1}{2.102}+\frac{1}{3.103}+...+\frac{1}{25.125}\)

\(A=\frac{1}{100}.\left(1-\frac{1}{101}\right)+\frac{1}{100}.\left(\frac{1}{2}-\frac{1}{102}\right)+\frac{1}{100}.\left(\frac{1}{3}-\frac{1}{103}\right)+...+\frac{1}{100}.\left(\frac{1}{25}-\frac{1}{125}\right)\)

\(A=\frac{1}{100}.\left(1-\frac{1}{101}+\frac{1}{2}-\frac{1}{102}+\frac{1}{3}-\frac{1}{103}+...+\frac{1}{25}-\frac{1}{125}\right)\)

\(A=\frac{1}{100}.\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{25}-\frac{1}{101}-\frac{1}{102}-\frac{1}{103}-...-\frac{1}{125}\right)\)

\(B=\frac{1}{1.26}+\frac{1}{2.27}+\frac{1}{3.28}+...+\frac{1}{100.125}\)

\(B=\frac{1}{25}.\left(1-\frac{1}{26}\right)+\frac{1}{25}.\left(\frac{1}{2}-\frac{1}{27}\right)+\frac{1}{25}.\left(\frac{1}{3}-\frac{1}{28}\right)+...+\frac{1}{25}.\left(\frac{1}{100}-\frac{1}{125}\right)\)

\(B=\frac{1}{25}.\left(1-\frac{1}{26}+\frac{1}{2}-\frac{1}{27}+\frac{1}{3}-\frac{1}{28}+...+\frac{1}{100}-\frac{1}{125}\right)\)

\(B=\frac{1}{25}.\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{100}-\frac{1}{26}-\frac{1}{27}-\frac{1}{28}-...-\frac{1}{125}\right)\)

\(B=\frac{1}{25}.\left(1+\frac{1}{2}+...+\frac{1}{25}+\frac{1}{26}+\frac{1}{27}+...+\frac{1}{100}-\frac{1}{26}-\frac{1}{27}-...-\frac{1}{100}-\frac{1}{101}-...-\frac{1}{125}\right)\)\(B=\frac{1}{25}.\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{25}-\frac{1}{101}-\frac{1}{102}-\frac{1}{103}-...-\frac{1}{125}\right)\)

Ta thấy biểu thức trong ngoặc của hai vế A và B giống nhau

Vậy A : B = \(\frac{1}{100}:\frac{1}{25}=\frac{1}{4}\)

\(A=\frac{1}{1.101}+\frac{1}{2.102}+\frac{1}{3.103}+...+\frac{1}{25.125}\)

\(\Rightarrow A=\left(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{24.25}\right)+\left(\frac{1}{101.102}+\frac{1}{102.103}+...+\frac{1}{124.125}\right)\)

\(A=\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{24}-\frac{1}{25}\right)+\left(\frac{1}{101}-\frac{1}{102}+\frac{1}{102}-\frac{1}{103}+...+\frac{1}{124}-\frac{1}{125}\right)\)

\(A=\left(1-\frac{1}{25}\right)+\left(\frac{1}{101}-\frac{1}{125}\right)\)

\(A=\frac{24}{25}+\frac{24}{12625}\)

Bạn tự tính luôn nha trog máy tính của mình là : 0,961... ( k làm thành phân số được )

A = 1 - 1/64 = 63/64 

tk nha

\(=1-\frac{1}{256}=\frac{255}{256}\)

Nhận xét :

1/2 = 1 - 1/2   ;   1/4 = 1/2 - 1/4   ;   1/8 = 1/4 - 1/8   ;   .....   ;   1/256 = 1/128 - 1/256

=> A = 1 - 1/2 + 1/2 - 1/4 + 1/4 - 1/8 + ..... + 1/128 - 1/256

=> A = 1 - 1/256 = 255/256

`@` `\text {Ans}`

`\downarrow`

\(\text{ A = }\dfrac{1}{4\times8}+\dfrac{1}{8\times12}+\dfrac{1}{12\times16}+...+\dfrac{1}{172\times176}\)

\(\text{A = }\dfrac{1}{4}\times\left(\dfrac{4}{4\times8}+\dfrac{4}{12\times16}+...+\dfrac{4}{172\times176}\right)\)

\(\text{A = }\dfrac{1}{4}\times\left(\dfrac{1}{4}-\dfrac{1}{8}+\dfrac{1}{12}-\dfrac{1}{16}+...+\dfrac{1}{172}-\dfrac{1}{176}\right)\)

\(\text{A = }\dfrac{1}{4}\times\left(\dfrac{1}{4}-\dfrac{1}{176}\right)\)

\(\text{A = }\dfrac{1}{4}\times\dfrac{43}{176}\)

\(\text{A = }\dfrac{43}{704}\)

Đáp số: `\text {A =} 43/704.`