\(A< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}\)

\(=1-\frac{1}{50}< 1< 2\Rightarrow A< 2\Rightarrowđpcm\)

thanks ban vi minh dang rat can dap an nay

\(A=\frac{1}{1^2}+\frac{1}{2^2}+.....+\frac{1}{50^2}\)

\(< 1+\frac{1}{1.2}+\frac{1}{2.3}+.....+\frac{1}{49.50}\)

\(=1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.....+\frac{1}{49}+\frac{1}{50}\)

\(=2-\frac{1}{50}< 2\)

Vậy A<2

mỗi p/số của A đều bé hơn 1/1.2+1/2.3+1/3.4+......+1/49.50

A<1-1/2+1/2-1/3+1/3-1/4+..........+1/49-1/50(tách ra thành hiệu)

A<1-1/50

mà 1/50>0=>1-1/50<1<2

A<1-1/50<1<2

A<2

chúc học tốt

\(A=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{49}-\frac{1}{50}\)
\(A=\left(1+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{49}\right)-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{50}\right)\)

\(A=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+...+\frac{1}{49}+\frac{1}{50}\right)-2.\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{50}\right)\)

\(A=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+...+\frac{1}{49}+\frac{1}{50}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{25}\right)\)

\(A=\frac{1}{26}+\frac{1}{27}+\frac{1}{28}+...+\frac{1}{50}\)

\(A=\left(\frac{1}{26}+\frac{1}{27}+...+\frac{1}{30}\right)+\left(\frac{1}{31}+\frac{1}{32}+...+\frac{1}{40}\right)+\left(\frac{1}{41}+\frac{1}{42}+...+\frac{1}{50}\right)\)

\(A< \left(\frac{1}{25}+\frac{1}{25}+...+\frac{1}{25}\right)+\left(\frac{1}{30}+\frac{1}{30}+...+\frac{1}{30}\right)+\left(\frac{1}{40}+\frac{1}{40}+...+\frac{1}{40}\right)\)

                5 phân số 1/25                                 10 phân số 1/30                                 10 phân số 1/40

\(A< 5.\frac{1}{25}+10.\frac{1}{30}+10.\frac{1}{40}\)

\(A< \frac{1}{5}+\frac{1}{3}+\frac{1}{4}\)

\(A< \frac{1}{4}+\frac{1}{3}+\frac{1}{4}\)

\(A< \frac{1}{2}+\frac{1}{3}\)

\(A< \frac{5}{6}\)

\(A=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}\)

\(=\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}\)

\(=\frac{5}{6}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{49}-\frac{1}{50}\)

\(=\frac{5}{6}+\left(\frac{1}{5}-\frac{1}{4}\right)+\left(\frac{1}{7}-\frac{1}{6}\right)+...+\left(\frac{1}{49}-\frac{1}{48}\right)-\frac{1}{50}\)

\(\frac{1}{5}-\frac{1}{4}< 0\)

\(\frac{1}{7}-\frac{1}{6}< 0\)

\(...\)

\(\frac{1}{49}-\frac{1}{48}< 0\)

\(\frac{5}{6}\) khi cộng với các số nhỏ hơn 0 thì giá trị nó sẽ giảm, đồng thời còn bớt đi \(\frac{1}{50}\)

Do đó \(A< \frac{5}{6}\)