a)\(\frac{8}{9}.\frac{15}{16}.\frac{24}{25}.....\frac{2499}{2500}\) \(=\frac{2.4}{3^2}.\frac{3.5}{4^2}.\frac{4.6}{5^2}.....\frac{49.51}{50^2}\)\(=\frac{\left(2.3.4.....49\right)\left(4.5.6....51\right)}{\left(3.4.5.....50\right)\left(3.4.5.....50\right)}=\frac{2.51}{50.3}=\frac{1.17}{25}=\frac{17}{25}\)

b)\(\left(1-\frac{1}{3}\right)\left(1-\frac{1}{6}\right)\left(1-\frac{1}{10}\right)......\left(1-\frac{1}{780}\right)\)

\(=\frac{2}{3}.\frac{5}{6}.\frac{9}{10}......\frac{779}{780}\)\(=\frac{4}{6}.\frac{10}{12}.\frac{18}{20}.....\frac{1558}{1560}\)

\(=\frac{1.4}{2.3}.\frac{2.5}{3.4}.\frac{3.6}{4.5}.....\frac{38.41}{39.40}\)

\(=\frac{\left(1.2.3.....38\right)\left(4.5.6.....41\right)}{\left(2.3.4.....39\right)\left(3.4.5.....40\right)}=\frac{1.41}{39.3}=\frac{41}{117}\)

\(\frac{4}{1}\)

B = 3/4.8/9.15/16.....2499/2500

= 1.3/2^2 . 2.4/3^2 . 3.5/4^2 ..... 49.51/50^2

= 1.2.3.....49/1.2.3.....50 . 3.4.5....51/1.2.3.....50

= 1/50 . 51/2

= 51/100

                                                                               Giải

B = 3/4.8/9.15/16.....2499/2500

B=1.3/2^2 . 2.4/3^2 . 3.5/4^2 ..... 49.51/50^2

B=1.2.3.....49/2.3.4.....50 . 3.4.5.....51/2.3.4.....50

B=1/50 . 51/2 

B=51/100

A=1.3/2.2x2.4/3.3x3.5/4.4x...29.31/30.30

A=(1.2.3....29)x(2.3.4.....31)/(2.3.4....30)x(2.3.4....30)

A=31/30

A=31/30

Trước hết ta đi chứng minh 1 bài toán \(a^2-1=\left(a-1\right)\left(a+1\right)\) (1)

Thật vậy \(\left(a-1\right)\left(a+1\right)=\left(a-1\right)a+\left(a-1\right)=a^2-a+a-1=a^2-1\)

Áp dụng (1) ta có:

\(A=\dfrac{2\cdot4}{3^2}\cdot\dfrac{3\cdot5}{4^2}\cdot...\cdot\dfrac{7\cdot9}{8^2}\\ =\dfrac{2\cdot3\cdot4^2\cdot...\cdot7^2\cdot8\cdot9}{3^2\cdot4^2\cdot...\cdot8^2}=\dfrac{2\cdot9}{3\cdot8}\\ =\dfrac{18}{24}=\dfrac{3}{4}\)

17/25 nha bạn