Question

[Nguyễn Anh Viên]

Cho A= 1/2^2 + 1/3^2 + .........+1/2012^2 + 1/2013^2

Chứng tỏ A<1

Các bạn giúp mình nhé !

1/2^2 chính là 1 phần 2 mũ 2 và tương tự nha các bạn.

Answer 1

[Nguyễn Thanh Hằng]

Ta có :

\(A=\dfrac{1}{2^2}+\dfrac{1}{3^2}+............+\dfrac{1}{2012^2}+\dfrac{1}{2013^2}\)

\(\Rightarrow A< \dfrac{1}{1.2}+\dfrac{1}{2.3}+...............+\dfrac{1}{2011.2012}+\dfrac{1}{2012.2013}\)

\(\Rightarrow A< \dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...........+\dfrac{1}{2011}-\dfrac{1}{2012}+\dfrac{1}{2012}-\dfrac{1}{2013}\)

\(\Rightarrow A< 1-\dfrac{1}{2013}< 1\)

\(\Rightarrow A< 1\rightarrowđpcm\)

Answer 2

[Sáng]

Lời giải:

Dễ thấy:

\(\dfrac{1}{2^2}=\dfrac{1}{2.2}< \dfrac{1}{1.2}\)

\(\dfrac{1}{3^2}=\dfrac{1}{3.3}< \dfrac{1}{2.3}\)

\(....\)

\(\dfrac{1}{2012^2}=\dfrac{1}{2012.2012}< \dfrac{1}{2011.2012}\)

\(\dfrac{1}{2013^2}=\dfrac{1}{2013.2013}< \dfrac{1}{2012.2013}\)

\(\Rightarrow A< \dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{2011.2012}+\dfrac{1}{2012.2013}\)

\(\Rightarrow A< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{2011}-\dfrac{1}{2012}+\dfrac{1}{2012}-\dfrac{1}{2013}\)

\(\Rightarrow A< 1-\dfrac{1}{2013}\)

\(\Rightarrow A< 1\left(dpcm\right)\)