A = 1/3^2 + 1/4^2 + 1/5^2 + ... + 1/50^2

1/3^2 = 1/9

1/4^2 < 1/3.4 = 1/3 - 1/4

1/5^2 < 1/4.5 = 1/4 - 1/5

.............................................

1/50^2 < 1/49.50 = 1/49 - 1/50

Cộng vế với vế ta có:

A = 1/3^2+1/4^2+..+1/50^2 = 1/9 + 1/3 - 1/50

A = 4/9 - 1/50 < 4/9

1/3^2 = 1/9

1/4^2 > 1/4.5 = 1/4 - 1/5

1/5^2 > 1/5.6 = 1/5 - 1/6

............................................

1/50^2 > 1/49.50 = 1/49 - 1/50

Cộng vế với vế ta có:

A = 1/3^2+1/4^2+ ...+ 1/50^2 > 1/9+1/4-1/50

A > 1/4 + (1/9 - 1/50)

1/9 > 1/50

1/9 - 1/50 > 0

A > 1/4 + 1/9 - 1/50 > 1/4

Vậy 1/4 < A < 4/9 (đpcm)

\(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}=\frac{1}{26}+\frac{1}{27}+\frac{1}{28}+...+\frac{1}{50}\)

\(\Rightarrow\left(1+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{49}\right)-\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{50}\right)=\frac{1}{26}+\frac{1}{27}+\frac{1}{28}+...+\frac{1}{50}\)

\(\Rightarrow\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{50}\right)-2\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{50}\right)=\frac{1}{26}+\frac{1}{27}+...+\frac{1}{50}\)

\(\Rightarrow\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{50}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{25}\right)=\frac{1}{26}+\frac{1}{27}+...+\frac{1}{50}\)

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

\(\Rightarrowđpcm\)

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

\(=\frac{1}{26}+\frac{1}{27}+...+\frac{1}{50}\left(\text{đ}pcm\right)\)

Chúc bạn học tốt !!!

Ta biến đổi vế phải :

1-1/2+1/3-1/4+.....+1/49-1/50

=(1+1/3+1/5+....+1/49)-(1/2+1/4+1/6+.......+1/50)

=(1+1/2+1/3+.....+1/49+1/50)-2(1/2+1/4+1/6+......+1/50)

=(1+1/2+...+1/50)-(1+1/2+1/3+....+1/25)

=1/26+1/27+.......+1/50

Vậy 1/26+1/27+1/28+.....+1/50=1-1/2+1/3-1/4+......+1/49-1/50

Mình không bấm phân số được mong mấy bạn thông cảm