Đặt \(A=\frac{1}{4^2}+\frac{1}{6^2}+...+\frac{1}{\left(2n\right)^2}\)

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

Có:

\(\frac{1}{2^2}< \frac{1}{1.2}\)

\(\frac{1}{3^2}< \frac{1}{2.3}\)

\(...\)

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

\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}< \frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{\left(n-1\right)n}\)

\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{n-1}-\frac{1}{n}\)

\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}< 1-\frac{1}{n}< 1\)

\(\Rightarrow A< \frac{1}{2^2}.1=\frac{1}{4}\)

Ta co =>A=1/2^2(1/2^2+1/2^3+1/2^4+...+1/n^2)

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

 =>1/(1.2)+1/(2.3)+...+1/(n-1).n=1/1-1/n=1-1/n

=>A=1/4(1-1/n)

Mà 1-1/n>1

=>A=1/4(1-1/n)<1/4.1+1/4

Vay A<1/4

bài này tui bít làm nhưng dài lắm

Cố gắng làm hộ mình với !

=>\(\frac{1}{2^2}\)x (\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+\(\frac{1}{4^2}\)+...+\(\frac{1}{n^2}\))

Đặt A=\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+\(\frac{1}{4^2}\)+...+\(\frac{1}{n^2}\)

Ta có:\(\frac{1}{2^2}\)<\(\frac{1}{1\cdot2}\)

         \(\frac{1}{3^2}\)<\(\frac{1}{2\cdot3}\)

.........\(\frac{1}{n^2}\)<\(\frac{1}{\left(n-1\right)\cdot n}\)

=>\(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+...+\(\frac{1}{n^2}\)<\(\frac{1}{1\cdot2}\)+\(\frac{1}{2\cdot3}\)+...+\(\frac{1}{\left(n-1\right)\cdot n}\)

=>A<1-\(\frac{1}{2}\)+\(\frac{1}{2}\)-\(\frac{1}{3}\)+...+\(\frac{1}{n-1}\)--\(\frac{1}{n}\)
=>\(\frac{1}{2^2}\)*A<\(\frac{1}{2^2}\)(1--\(\frac{1}{n}\))

=>\(\frac{1}{2^2}\)*A<\(\frac{1}{4}\)(1--\(\frac{1}{n}\))

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

=>\(\frac{1}{2^2}\)+\(\frac{1}{4^2}\)+...+\(\frac{1}{\left(2n\right)^2}\)<\(\frac{1}{4}\)

đặt A=1/2^2+1/4^2+1/6^2+.....+1/(2n)^2

ta có :

A=1/2^2 +1/2^2(1/2^2+1/3^2+1/4^2+.....+1/n^2)

A<1/2^2+1/2^2(1/1.2+1/2.3+...+1/(n-1)n)

=1/2^2+1/2^2(1-1/2+1/2-1/3+....+1/(n-1)-1/n)

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

<1/2^2+1/2^2.1=1/2<3/4

vậy A<3/4

mình đồng ý với bạn witch roses

ta có 

\(\frac{1}{2^2}

1/ Ta có:

\(a^5-a^3+a=2\)

Dễ thấy a = 0 không phải là nghiệm từ đó ta có:

\(a^6-a^4+a^2=2a\)

\(\Rightarrow2a=a^6+a^2-a^4\ge2a^4-a^4\ge a^4\)

\(\Rightarrow\hept{\begin{cases}2a\ge a^4\\a>0\end{cases}}\)

\(\Leftrightarrow\hept{\begin{cases}2\ge a^3\\a>0\end{cases}}\)

\(\Leftrightarrow\hept{\begin{cases}4\ge a^6\\a>0\end{cases}}\)

Dấu = không xảy ra 

Vậy \(a^6< 4\)

Câu 2/

Câu hỏi của XPer Miner - Toán lớp 9 - Học toán với OnlineMath

Bạn tham khảo cách làm của bạn Alibabba nguyễn nha!!

Ta có:

\(M=\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+...+\frac{1}{\left(2n\right)^2}\)

\(=\frac{1}{2^2.2^2}+\frac{1}{2^2.3^2}+\frac{1}{2^2.4^2}+...+\frac{1}{2^2.n^2}\)

\(=\frac{1}{2^2}.\frac{1}{2^2}+\frac{1}{2^2}.\frac{1}{3^2}+\frac{1}{2^2}.\frac{1}{4^2}+...+\frac{1}{2^2}.\frac{1}{n^2}\)

\(=\frac{1}{2^2}.\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{n^2}\right)\)

\(=\frac{1}{4}.\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{n^2}\right)\)

Coi \(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{n^2}\)

\(=\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+...+\frac{1}{n.n}\)

Vì: \(\frac{1}{2.2}

kho qua minh moi lop 5

What? lớp 5 mà học cả số mũ?? thời nay bọn trẻ con học trâu thật!

a. Ta có: \(\frac{1}{2^2}\)< \(\frac{1}{1.3}\)

\(\frac{1}{4^2}\)< 1/(3.5)

1/(6^2) <1/(5.7)

...

1/(2n)^2 < 1/(2n-1)(2n+1)

=> 1/2^2 +1/4^2 + 1/6^2 +...+1/(2n)^2 < 1/(1.3) +...+1/(2n-1)(2n+1)

=> 2(1/2^2 +1/4^2 + 1/6^2 +...+1/(2n)^2) < (1/1 - 1/3 +1/3 - 1/5 + 1/5 - 1/7 +...+ 1/(2n-1) - 1/(2n+1)

=>2(1/2^2 +1/4^2 + 1/6^2 +...+1/(2n)^2) < 1 - 1/(2n+1) = 2n/(2n+1)

=> 1/2^2 +1/4^2 + 1/6^2 +...+1/(2n)^2 < 2n/(2n+1) . 1/2

Vì 2n/2n+1 < 1 =>  2n/(2n+1) . 1/2 < 1/2

=> 1/2^2 +1/4^2 + 1/6^2 +...+1/(2n)^2 <1/2

 Câu b tương tự

\(\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+\).... \(+\frac{1}{\left(2n\right)^2}\)= \(\frac{1}{2^2}\). ( \(\frac{1}{2^2}+\frac{1}{3^2}+.....+\frac{1}{n^2}\)) < \(\frac{1}{2^2}\)( \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{\left(n-1\right).\left(n\right)}\)) = \(\frac{1}{2^2}\)( \(1-\frac{1}{n}\)) < \(\frac{1}{2^2}\).1 = \(\frac{1}{4}\)

\(\Rightarrow\)\(\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+...+\frac{1}{\left(2n\right)^2}\)< \(\frac{1}{4}\)

mình ko hiểu lắm