Chứng minh rằng:\(\dfrac{1}{3^2}\)+ \(\dfrac{1}{4^2}\)+\(\dfrac{1}{5^2}\) + \(\dfrac{1}{6^2}\)+...+\(\dfrac{1}{100^2}\) <\(\dfrac{1}{2}\) 
Mình cần gấp!

Tính: A= 4.\(5^{100}\)(\(\dfrac{1}{5}\)+\(\dfrac{1}{5^2}\)+\(\dfrac{1}{5^3}\)+...+\(\dfrac{1}{5^{99}}\)+\(\dfrac{1}{5^{100}}\))+1

Tính nhanh :

1\(\dfrac{4}{5}\) + 2\(\dfrac{5}{7}\) + 3\(\dfrac{4}{5}\) + 4\(\dfrac{5}{7}\)

Giúp mik nhanh với mik đang cần gấp

Tính giá trị biểu thức :

\(\dfrac{1}{2}-\left(\dfrac{1}{3}+\dfrac{2}{3}\right)+\left(\dfrac{1}{4}+\dfrac{2}{4}+\dfrac{3}{4}\right)-\left(\dfrac{1}{5}+\dfrac{2}{5}+\dfrac{3}{5}+\dfrac{4}{5}\right)+\left(\dfrac{1}{6}+\dfrac{2}{6}+\dfrac{3}{6}+\dfrac{4}{6}+\dfrac{5}{6}\right)-\left(\dfrac{1}{7}+\dfrac{2}{7}+\dfrac{3}{7}+\dfrac{4}{7}+\dfrac{5}{7}+\dfrac{6}{7}\right)+...+\left(100+...+\dfrac{99}{100}\right)\)

Rút gọn ;

D = \(\dfrac{1}{5}-\dfrac{1}{5^2}+\dfrac{1}{5^3}-\dfrac{1}{5^4}+\dfrac{1}{5^5}-\dfrac{1}{5^6}+...+\dfrac{1}{5^{99}}-\dfrac{1}{5^{100}}+\dfrac{1}{6.5^{100}}\)

 Cho biểu thức \(A=-\dfrac{1}{5}+\dfrac{1}{5^2}-\dfrac{1}{5^3}+\dfrac{1}{5^4}-\dfrac{1}{5^5}+...+\dfrac{1}{5^{100}}\)

Cho \(S=\dfrac{1}{5^2}+\dfrac{2}{5^3}+\dfrac{3}{5^4}+...+\dfrac{99}{5^{100}}\). Chứng tỏ rằng S<\(\dfrac{1}{16}\)

Cho S=\(\dfrac{1}{5^2}+\dfrac{2}{5^3}+\dfrac{3}{5^4}+...+\dfrac{99}{5^{100}}\) . Chứng tỏ rằng \(S< \dfrac{1}{16}\)

Cho \(A=1+\dfrac{3}{2^3}+\dfrac{4}{2^4}+\dfrac{5}{2^5}+...+\dfrac{100}{2^{100}}\). Chứng minh A < 2.