Question

[Nguyễn Thị Bích Hà]

Chứng minh rằng

a,M= 1/2² +1/3² +1/4² +......+1/n² <1(1 thuộc n)

b, N=1/4²+1/6²+1/8²+........+1/(2n)²<1/4

c, Cho sáu số nguyên a<b<c<d<m<n

Chứng minh rằng a+c+m/a+b+c+d+m+n<1/2

Giúp mình với nhé mai mình phải nộp rồi

Answer 1

[JakiNatsumi]

\(a,M=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{n^2}\)

\(M< \dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{\left(n-1\right)n}\)

\(M< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{n-1}-\dfrac{1}{n}\)

\(M< 1-\dfrac{1}{n}< 1\)

\(\Rightarrow M< 1\left(đpcm\right)\)

\(b,N=\dfrac{1}{4^2}+\dfrac{1}{6^6}+\dfrac{1}{8^2}+...+\dfrac{1}{\left(2n\right)^2}\)

\(N< \dfrac{1}{3.5}+\dfrac{1}{5.7}+\dfrac{1}{7.9}+...+\dfrac{1}{\left(2n-1\right)\left(2n+1\right)}\)

\(N< \dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+...+\dfrac{1}{2n-1}-\dfrac{1}{2n+1}\)

\(N< \dfrac{1}{3}-\dfrac{1}{2n+1}< \dfrac{1}{3}\)

\(c,\) Vì \(a< b\Rightarrow2a< a+b\)

\(c< d\Rightarrow2c< c+d\)

\(m< n\Rightarrow2m< m+n\)

\(\Rightarrow2a+2c+2m=2.\left(a+c+m\right)< a+b+c+d+m+n\)

\(\Rightarrow\dfrac{a+c+m}{a+b+c+d+m}< \dfrac{1}{2}\)