Tìm $x, y$ biết $\dfrac{4^x}{2^{x+y}}=8$ và $\dfrac{9^{x+y}}{3^{5 y}}=243$ $(x, y \in \mathbb{N}) .$

So sánh $5^{165}$ và $7^{110}$.

Tìm $x, y, z$ biết: $(3 x-5)^{2\,022}+\Big(y^2-\dfrac{4}{9}\Big)^{2\,024}+(x-2 z)^{2\,026}=0 .$

Tìm $x$ biết: $\dfrac{x+3}{400}+\dfrac{x+4}{399}+\dfrac{x+5}{398}+\dfrac{x+6}{397}+\dfrac{x+7}{396}+\dfrac{x+433}{6}=0$.

Cho $1^2+2^2+3^2+\ldots+10^2=385$.

Tinh $S=\left(12^2+14^2+16^2+18^2+20^2\right)-\left(1^2+3^2+5^2+7^2+9^2\right)$.

So sánh $A$ và $B$ biết: $A=\dfrac{2\,022^{2\,022}+1}{2\,022^{2\,023}+1}$ và $B=\dfrac{2\,022^{2\,021}+1}{2\,022^{2\,022}+1}$.

Tìm số nguyên $x$ thỏa mãn: $\dfrac{2\,024}{x(x+2)}+\dfrac{2\,024}{(x+2)(x+4)}+\dfrac{2\,024}{(x+4)(x+6)}-\dfrac{1\,012}{x}=\dfrac{1}{2}$ với $x \neq \{0; -2; -4; -6\}$.

So sánh $A=\dfrac{7^{2\,011}+2}{7^{2\,012}+2}$ và $B=\dfrac{7^{2\,023}+2}{7^{2\,024}+2}$.

Tính $A=\dfrac{1}{-2} \cdot \dfrac{1}{3}+\dfrac{1}{-3} \cdot \dfrac{1}{4}+\ldots+\dfrac{1}{-9} \cdot \dfrac{1}{10}$.

Tinh nhanh $M=\dfrac{4}{11.16}+\dfrac{4}{16\cdot 21}+\dfrac{4}{21\cdot 26}+\ldots+\dfrac{4}{61\cdot 66}$.

Cho $M=\dfrac{1}{7^2}-\dfrac{1}{7^4}+\dfrac{1}{7^6}-\dfrac{1}{7^8}+ \ldots+\dfrac{1}{7^{98}}-\dfrac{1}{7^{100}}$. Chứng minh rằng $M\lt \dfrac{1}{50}$.

Tìm $x$, biết:

$x-\dfrac{5}{11}=\dfrac{1}{3}-\dfrac{1}{12}-\dfrac{1}{20}-\dfrac{1}{30}-\dfrac{1}{42} - \ldots-\dfrac{1}{90}-\dfrac{1}{110}$

Tính $B=1+\dfrac{1}{13}+\dfrac{1}{13^2}+\dfrac{1}{13^3}+\ldots +\dfrac{1}{13^n}\left(n \in \mathbb{N}^*\right) .$

Tính giá trị biểu thức: $A=\Big(\dfrac{3}{2}-\dfrac{2}{2^2}\Big)\Big(\dfrac{4}{3}-\dfrac{2}{3^2}\Big)\Big(\dfrac{5}{4}-\dfrac{2}{4^2}\Big) \cdots\Big(\dfrac{101}{100}-\dfrac{2}{100^2}\Big)$.

Cho $A=\dfrac{1}{3^2}+\dfrac{1}{5^2}+\dfrac{1}{7^2}+\ldots+\dfrac{1}{2\,025^2}$. Chứng minh rằng $A\lt \dfrac{1}{4}$.

So sánh $P=\dfrac{1}{2^2}+\dfrac{1}{2^4}+\dfrac{1}{2^6}+\ldots +\dfrac{1}{2^{2\,020}}+\dfrac{1}{2^{2\,022}}$ với $\dfrac{1}{3}$.

Cho biểu thức

$M = \dfrac{1}{5}+\dfrac{1}{5^2}+\dfrac{1}{5^3}+\ldots+\dfrac{1}{5^{2\,024}}+\dfrac{1}{5^{2\,025}}$. Chứng minh rằng: $M\lt \dfrac{1}{2}$.

Chứng minh rằng: $S=\dfrac{1}{5^2}+\dfrac{2}{5^3}+\dfrac{3}{5^4}+\ldots+\dfrac{99}{5^{100}}\lt \dfrac{1}{16}$

Cho $A=\dfrac{1}{7}+\dfrac{1}{7^2}+\dfrac{1}{7^3}+\dfrac{1}{7^4}+\ldots +\dfrac{1}{7^{2\,023}}+\dfrac{1}{7^{2\,024}}$.

Tìm $x$ để $6A=1-\dfrac1{7^x}$.

Tìm $x$, biết: $\Big(x+\dfrac{1}{2}\Big)^{2\,024}=2-\Big(1+\dfrac{1}{2}+\dfrac{1}{2^2}+\ldots+\dfrac{1}{2^{2\,023}}+\dfrac{1}{2^{2\,024}}\Big)$.