Question

CHứng minh rằng \(\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{50^2}< 1\)

Answer 1

Đặt:

\(A=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{50^2}\)

\(\left\{{}\begin{matrix}\dfrac{1}{2^2}< \dfrac{1}{1.2}\\\dfrac{1}{3^2}< \dfrac{1}{2.3}\\\dfrac{1}{4^2}< \dfrac{1}{3.4}\\\dfrac{1}{50^2}< \dfrac{1}{49.50}\end{matrix}\right.\)

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

\(A< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{49}-\dfrac{1}{50}\)

\(A< 1-\dfrac{1}{50}\)

\(A< 1\rightarrowđpcm\)

Answer 2

Ta thấy:

\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{50^2}< \dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{49.50}\)

\(\Leftrightarrow\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{50^2}< 1-\dfrac{1}{50}\)

\(\Leftrightarrow\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{50^2}< \dfrac{49}{50}< 1\)

Vậy \(\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{50^2}< 1\)

Answer 3

Đặt :

\(A=\dfrac{1}{2^2}+\dfrac{1}{3^2}+................+\dfrac{1}{50^2}\)

Ta thấy :

\(\dfrac{1}{2^2}=\dfrac{1}{1.2}\)

\(\dfrac{1}{3^2}=\dfrac{1}{2.3}\)

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

\(\dfrac{1}{50^2}< \dfrac{1}{49.50}\)

\(\Leftrightarrow A< \dfrac{1}{1.2}+\dfrac{1}{2.3}+.............+\dfrac{1}{49.50}\)

\(\Leftrightarrow A< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...........+\dfrac{1}{49}-\dfrac{1}{50}\)

\(\Leftrightarrow A< 1-\dfrac{1}{50}< 1\)

\(\Leftrightarrow A< 1\)

Vậy ................