Question

Cho A=\(\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+\dfrac{1}{3.4.5}+...+\dfrac{1}{2014.2015.2016}\).So sánh A với \(\dfrac{1}{4}\)

Giúp mình nha!!

Answer 1

\(A=\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+\dfrac{1}{3.4.5}+...+\dfrac{1}{2014.2015.2016}\)

\(A=\dfrac{1}{2}\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{2014.2015}+\dfrac{1}{2015.2016}\right)\)

\(A=\dfrac{1}{2}\left(\dfrac{1}{1.2}-\dfrac{1}{2015.2016}\right)=\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{2015.2016}\right)\)

\(A=\dfrac{1}{4}-\dfrac{1}{2.2015.2016}< \dfrac{1}{4}\)

\(=>A< \dfrac{1}{4}\)

Chúc bn học tốt

Answer 2

Ta có \(\dfrac{1}{1.2}-\dfrac{1}{2.3}=\dfrac{2}{1.2.3};\dfrac{1}{2.3}-\dfrac{1}{3.4}=\dfrac{2}{2.3.4};\dfrac{2}{3.4.5};...;\dfrac{1}{2014.2015}-\dfrac{1}{2015.2016}=\dfrac{2}{2014.2015.2016}\)

2A= \(\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+\dfrac{2}{3.4.5}+...+\dfrac{2}{2014.2015.2016}\)

2A=\(\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{3.4}-\dfrac{1}{4.5}+\dfrac{1}{5.6}-\dfrac{1}{6.7}+...+\dfrac{1}{2014.2015}-\dfrac{1}{2015.2016}\)

2A=\(\dfrac{1}{1.2}-\dfrac{1}{2015.2016}\)

A=\(\left(\dfrac{1}{2}:2\right)-\left(\dfrac{1}{2015.2016}:2\right)\)

A= \(\dfrac{1}{4}-\dfrac{1}{2015.2016.2}< \dfrac{1}{4}\)

Vậy A<\(\dfrac{1}{4}\)