Đề là gì bạn nhỉ?

Đề bài yêu cầu gì?

1, Thấy : \(\frac{1}{5}< \frac{2}{2.4}\)

                \(\frac{1}{13}< \frac{2}{4.6}\)

                  .....

                  \(\frac{1}{n^2+\left(n+1\right)^2}< \frac{2}{2n\left(2n+1\right)}\)

Cộng từng vế có :

 \(\frac{1}{5}+\frac{1}{13}+...+\frac{1}{n^2+\left(n+1\right)^2}< \frac{2}{2.4}+\frac{2}{4.6}+...+\frac{2}{2n\left(2n+2\right)}\)

\(\frac{1}{5}+\frac{1}{13}+...+\frac{1}{n^2+\left(n+1\right)^2}< \frac{1}{2}-\frac{1}{4}+....+\frac{1}{2n}-\frac{1}{2n+2}\)

 \(\frac{1}{5}+\frac{1}{13}+..+\frac{1}{n^2+\left(n+1\right)^2}< \frac{1}{2}-\frac{1}{2n+2}\)

Mà \(\frac{1}{2}-\frac{1}{2n+2}< \frac{1}{2}\)=> Tổng trên < 1/2

2,M = \(\frac{3}{\left(1.2\right)^2}+\frac{5}{\left(2.3\right)^2}+...+\frac{2n+1}{\left[n\left(n+1\right)\right]^2}\)

=> M \(=\frac{1}{1^2}-\frac{1}{2^2}+\frac{1}{2^2}-\frac{1}{3^2}+...+\frac{1}{\left(n-1\right)^2}-\frac{1}{n^2}+\frac{1}{n^2}-\frac{1}{\left(n+1\right)^2}\)

    \(M=1-\frac{1}{\left(n+1\right)^2}=\frac{\left(n+1\right)^2-1}{\left(n+1\right)^2}=\frac{n^2+2n+1-1}{\left(n+1\right)^2}=\frac{n^2+2n}{\left(n+1\right)^2}\)

Đến đây tắc r tự nghĩ tiếp >:

tại xao 1/n^2+(n+1)^2 < 2n(2n+1)?

A  =(-1)²ⁿ.(-1)ⁿ.(-1)ⁿ⁺¹

A = (-1)^2n+n+n+1

A = (- 1)^{4n + 1}

A = (-1)⁴ⁿ.(-1)

A = 1.(-1)

A = -1 

1²+2²+3²+...+n²
=1.(2−1)+2.(3−1)+3.(4−1)+...+n[(n+1)−1]
=[1.2+2.3+3.4+...+n(n+1)]-(1+2+3+...+n)
=[n(n+1)(n+2)-0.1.2]/3-n(n+1)/2
=n(n+1)(2n+1)/6

\(A=\frac{1}{5}+\frac{1}{13}+\frac{1}{25}+...+\frac{1}{2.n^2+2n+1}< \frac{1}{4}+\frac{1}{12}+\frac{1}{24}+...+\frac{1}{2.n^2+2n}\)

\(A< \frac{1}{2}.\left(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...+\frac{1}{n.\left(n+1\right)}\right)\)

\(A< \frac{1}{2}.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{n.\left(n+1\right)}\right)\)

\(A< \frac{1}{2}.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{n}-\frac{1}{n+1}\right)\)

\(A< \frac{1}{2}.\left(1-\frac{1}{n+1}\right)< \frac{1}{2}\)

=> \(A< \frac{1}{2}\)

Câu 1

x-(-25-17-x)=6+x

<=>x+25+17+x=6+x

<=>2x-x=6-25-17

<=>x=-36

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