\(\dfrac{3}{2^2}.\dfrac{8}{3^2}.\dfrac{15}{4^2}.....\dfrac{899}{30^2}\)

= \(\dfrac{1.3}{2.2}.\dfrac{2.4}{3.3}.\dfrac{3.5}{4.4}.....\dfrac{29.31}{30.30}\) 

= \(\dfrac{1.3.2.4.3.5.....29.31}{2.2.3.3.4.4.....30.30}\)

=\(\dfrac{\left(1.2.3.....29\right).\left(3.4.5......31\right)}{\left(2.3.4......30\right).\left(2.3.4.....30\right)}\) 

= \(\dfrac{1.31}{2.30}\) 

= \(\dfrac{31}{60}\)

Ta có: \(\dfrac{3}{2^2}\cdot\dfrac{8}{3^2}\cdot\dfrac{15}{4^2}\cdot...\cdot\dfrac{899}{30^2}\)

\(=\dfrac{1\cdot3}{2\cdot2}\cdot\dfrac{2\cdot2^2}{3\cdot3}\cdot\dfrac{3\cdot5}{4\cdot4}\cdot...\cdot\dfrac{29\cdot31}{30\cdot30}\)

\(=\dfrac{\left(1\cdot2\cdot3\cdot...\cdot29\right)\cdot\left(3\cdot4\cdot5\cdot...\cdot31\right)}{\left(2\cdot3\cdot4\cdot...\cdot30\right)\left(2\cdot3\cdot4\cdot...\cdot30\right)}\)

\(=\dfrac{1\cdot31}{2\cdot30}=\dfrac{31}{60}\)

Đặt \(A=\frac{3}{2^2}.\frac{8}{3^2}.\frac{15}{4^2}.......\frac{899}{30^2}\)

\(A=\frac{1.3}{2^2}.\frac{2.4}{3^2}.\frac{3.5}{4^2}........\frac{29.31}{30^2}\)

\(A=\frac{\left(1.3\right).\left(2.4\right).\left(3.5\right).....\left(29.31\right)}{2^2.3^2.4^2.....30^2}\)

Để dễ rút gọn,ta viêt tử số của A dưới dạng tích các số tự nhiên liên tiếp:

\(A=\frac{\left(1.2.3.......29.29\right).\left(3.4.5.........30.31\right)}{\left(2.2\right).\left(3.3\right).\left(4.4\right)........\left(30.30\right)}\)

\(A=\frac{1.2.3.......28.29}{2.3.4.........29.30}.\frac{3.4.5.....30.31}{2.3.4.......29.30}=\frac{1}{30}.\frac{31}{2}=\frac{31}{60}\)

Vậy A=31/60

dài lắm

GIẢI RA THÌ DÀI LẮM!

Mỏi tay lam   ^(oo)^

\(Q=\dfrac{3}{2^2}.\dfrac{8}{3^2}.\dfrac{15}{4^2}.......................\dfrac{899}{30^2}\)

\(\Leftrightarrow Q=\dfrac{1.3.2.4.3.5........29.31}{2.2.3.3......30.30}\)

\(\Leftrightarrow Q=\dfrac{\left(2.3.....29.30\right)\left(3.4.5....29.31\right)}{\left(2.3.4....30\right)\left(2.3.4....30\right)}\)

\(\Leftrightarrow Q=\dfrac{31}{2.30}\)

\(\Leftrightarrow Q=\dfrac{31}{60}\)

A=\(\dfrac{3}{2^2}\)x\(\dfrac{8}{3^2}\)x\(\dfrac{15}{4^2}\)......\(\dfrac{899}{30^2}\)=\(\dfrac{1.3.2.4.3.5....29.31}{2.2.3.3.4.4....30.30}\)= (\(\dfrac{2.3....29.30}{2.3....20.30}\)).(\(\dfrac{3.4.5....29.31}{2.3.4....29.30}\))=\(\dfrac{31}{2.30}\)=\(\dfrac{31}{60}\)

3/2² x8/3² x15/4²x...x899/30²

=1.3/2.2x2.4/3.3x...x29.31/30.30 xong rồi gạch đi còn

=1/2x31/30=31/60