\(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+.....+\frac{1}{10000}=\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+.....+\frac{1}{100.100}\)

\(\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+....+\frac{1}{100.100}<\frac{1}{1.2}+\frac{1}{2.3}+.....+\frac{1}{99.100}=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-....-\frac{1}{99}+\frac{1}{99}-\frac{1}{100}\)\(=1-\frac{1}{100}=\frac{99}{100}<1\)

Vậy \(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+.....+\frac{1}{10000}<1\)

  zxzXZda

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\(A=\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+...+\frac{1}{100^2}\)

\(A< \frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{99.100}\)

\(A< \frac{3-2}{2.3}+\frac{4-3}{3.4}+\frac{5-4}{4.5}+\frac{6-5}{5.6}+...+\frac{100-99}{99.100}\)

\(A< \frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{100}\)

\(A< \frac{1}{2}-\frac{1}{100}=\frac{49}{100}=\left(\frac{7}{10}\right)^2\)

Ta có \(\frac{25}{36}=\left(\frac{5}{6}\right)^2\)

Ta thấy \(\frac{5}{6}=\frac{25}{30}>\frac{7}{10}=\frac{21}{30}\Rightarrow\left(\frac{7}{10}\right)^2< \left(\frac{5}{6}\right)^2\Rightarrow A< \left(\frac{7}{10}\right)^2< \left(\frac{5}{6}\right)^2=\frac{25}{36}\)

Ta có: 1 + 4 + 9 + 16 + 25 + ... + 10000

= 1 + 2² + 3² + 4² + 5² + ... + 100²

= 1.1 + 2.2. + 3.3 + 4.4 + 5.5 + ... + 100.100

= 1(2 - 1) + 2(3 - 1) + 3(4 - 1) + ... + 100(101 - 1)

= 1.2 - 1 + 2.3 - 2 + 3.4 - 3 + ... + 100.101 - 100

= (1.2 + 2.3 + 3.4 + ... + 100.101) - (1 + 2 + 3 + 4 + ... + 100)

Ta có: 3(1.2 + 2.3 + 3.4 + ... + 100.101) = 1.2.3 + 2.3.3 + 3.3.4 + ... + 3.100.101

= 1.2.(3 - 0) + 2.3.(4 - 1) + 3.4.(5 - 2) + ... + 100.101.(102 - 99)

= 1.2.3 - 1.2.0 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ... + 100.101.102 - 100.101.99

= 100.101.102

\(\Rightarrow\) 1.2 + 2.3 + 3.4 + ... + 100.101 = 100.101.34 = 343400

Ta có: 1 + 2 + 3 + 4 + ... + 100 = (100 + 1).100 : 2 = 5050

\(\Rightarrow\) (1.2 + 2.3 + 3.4 + ... + 100.101) - (1 + 2 + 3 + 4 + ... + 100) = 343400 - 5050 = 338350

Vậy 1 + 4 + 9 + 16 + 25 + ... + 10000 = 338350

Chúc bn học tốt!

số phần tử là:

(10000-1):3+1=3334

số số hạng là:

(10000+1).3334:2=16671667

Câu trả lời nè bạn:

 385

Bạn ơi số cuối cùng của dãy số là 10000 chứ ko phải là 100 bạn nhé

A=1/(2x2)+1/(3x3)+...+1/(100x100) 
Nhận thấy rằng n x n -1=n x n -n+n-1=n x (n-1)+n-1=(n-1) x (n+1) 
=> A < 1/(2x2-1)+1/(3x3-1)+...+1/(100x100-1)=1/(1x3)+1/(3x5)+...+1/(99x101)=1/2-1/202<1/2<3/4

A=1/(2x2)+1/(3x3)+...+1/(100x100) Nhận thấy rằng n x n -1=n x n -n+n-1=n x (n-1)+n-1=(n-1) x (n+1) => A < 1/(2x2-1)+1/(3x3-1)+...+1/(100x100-1)=1/(1x3)+1/(3x5)+...+1/(99x101)=1/2-1/202<1/2<3/4