A=\(\frac{3}{4}.\frac{8}{9}.\frac{15}{16}.........\frac{899}{900}\)

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

A=\(\frac{1.2.3.......29}{2.3.4.......30}.\frac{3.4.5........31}{2.3.4.......30}\)

A=\(\frac{1}{30}.\frac{2}{31}=\frac{1}{465}\)

A =\(\frac{2^2-1}{2^2}\)+ \(\frac{3^2-1}{3^2}\)+ \(\frac{4^2-1}{4^2}\)+,,,+

   = 1 - \(\frac{1}{2^2}\)+ 1 - \(\frac{1}{3^2}\)+ ...+ 1 - \(\frac{1}{30^2}\)

   = ( 1+ 1+1 +... + 1 ) - ( \(\frac{1}{2^2}\)+ \(\frac{1}{3^2}\)+ ... +\(\frac{1}{30^2}\))

   = 29 - ( \(\frac{1}{2^2}\)+ \(\frac{1}{3^2}\)+ ... +\(\frac{1}{30^2}\))

Vậy A không là số nguyên

\(A=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}....\frac{899}{900}\)

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

\(=\frac{1.2.3....29}{2.3.4....30}.\frac{3.4.5....31}{2.3.4....30}\)

\(=\frac{1}{30}.\frac{31}{2}=\frac{31}{60}\)

\(A=\frac{1\cdot3}{2\cdot2}\cdot\frac{2\cdot4}{3\cdot3}\cdot...\cdot\frac{29\cdot31}{30\cdot30}\)

\(A=\frac{1\cdot2\cdot...\cdot29}{2\cdot3\cdot...\cdot30}\cdot\frac{3\cdot4\cdot...\cdot31}{2\cdot3\cdot...\cdot30}=\frac{1}{30}\cdot\frac{31}{2}=\frac{31}{60}\)

A=3/4.8/9 .15/16.....899/900

A=1.3/2^2 .   2.4 /3^2  .   3.5/4^2 ....... 29.31 / 30^2

A= 1.2.3.....29 / 2.3.4....30   .   3.4.5...31 / 2.3.4....30

A=1/30 . 31/2 

A= 31/60

Nhớ k nha

A=\(\frac{31}{60}\)

\(A=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}.......\frac{899}{900}=\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}......\frac{29.31}{30.30}=\frac{1.2.3.....29}{2.3.4......30}.\frac{3.4.5......31}{2.3.4......30}\)

\(=\frac{1}{30}.\frac{31}{2}=\frac{31}{60}\)

chứng tỏ rằng 1/5 + 1/6 + 1/7 + .....+ 1/17 < 2 

Ta có: A=\(\frac{3}{4}.\frac{8}{9}.\frac{15}{16}.\frac{24}{25}....\frac{899}{900}\)

\(\Leftrightarrow A=\frac{1.3}{2^2}.\frac{2.4}{3^2}.\frac{3.5}{4^2}.\frac{4.6}{5^2}....\frac{29.31}{30^2}\)

\(\Leftrightarrow A=\frac{1.2.3.4...29}{2.3.4.5...30}.\frac{3.4.5.6...31}{2.3.4.5...30}\)

\(\Leftrightarrow A=\frac{1}{30}.\frac{31}{2}\)

\(\Leftrightarrow A=\frac{1.31}{30.2}\)

\(\Leftrightarrow A=\frac{31}{60}\)

\(A=\frac{3}{4}\times\frac{8}{9}\times\frac{15}{16}\times\frac{24}{25}\times...\times\frac{899}{900}\)

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

\(=\frac{\left(1\times2\times3\times...\times29\right)\left(3\times4\times5\times...\times31\right)}{\left(2\times3\times4\times...\times30\right)\left(2\times3\times4\times...\times30\right)}\)

\(=\frac{1\times2\times3\times...\times29}{2\times3\times4\times...\times30}.\frac{3\times4\times5\times...\times31}{2\times3\times4\times...\times30}\)

\(=\frac{1}{30}.\frac{31}{2}\)

\(=\frac{31}{60}\)

\(A=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}....\frac{899}{900}\\ =\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}....\frac{29.31}{30.30}\\ =\frac{1.2.3.4....29}{2.3.4...30}.\frac{3.4.5...31}{2.3.4...30}\\ =\frac{1}{30}.\frac{31}{2}=\frac{31}{60}\)

.

\(A=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}.\frac{24}{25}....\frac{899}{900}\)

\(A=\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}.\frac{4.6}{5.5}....\frac{29.31}{30.30}\)

\(A=\frac{1.2.3.4....29}{2.3.4....30}.\frac{3.4.5.6...31}{2.3.4...30}\)

\(A=\frac{1}{30}.\frac{31}{2}\) (Rút gọn theo chiều /// và \\\ nhé)

\(A=\frac{31}{60}\)

Chúc học tốt!~~