Question

Cho A= 1/1²+1/3²+1/42+...+1/50². Chứng minh A<2

Answer 1

Ta có:

1/3^2 < 1/2.3

1/4^2 < 1/3.4

....

1/50^2 < 1/49.50

=> A = 1/1^2 + 1/3^2+1/4^2+...+1/50^2 < 1 + 1/2.3 +1/3.4+...+1/49.50 = 1 + 1/2 - 1/3 + 1/3-1/4 + ...+1/49-1/50 = 1+1/2 - 1/50 < 2

Vậy A<2 (ĐPCM)

Answer 2

ta có:

1/1²=1

=> A>1

ta cần chứng minh

A-1<1

ta có

\(\frac{1}{3^2}<\frac{1}{2.3};\frac{1}{4^2}<\frac{1}{3.4};....;\frac{1}{50^2}<\frac{1}{49.50}\)\(\Rightarrow\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{50^2}<\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{49.50}=\frac{48}{100}<1\)

=> A-1<1

=>A<2