Sẵn sàng có 7 chữ cái

Vậy chữ cái thứ 1000 là :

\(1000:7=142\left(dư6\right)\)

Vậy chữ cái thứ 1000 là chữ n

\(S=3+3^2+3^3+...+3^{44}+3^{45}\)

\(3S=3.\left(3+3^2+3^3+...+3^{44}+3^{45}\right)\)

\(3S=3^2+3^3+3^4+...+3^{45}+3^{46}\)

\(3S-S=\left(3^2+3^3+3^4+...+3^{45}+3^{46}\right)-\left(3+ 3^2+3^3+...+3^{44}+3^{45}\right)\)

\(2S=3^{46}-3\)

\(S=\dfrac{3^{46}-3}{2}\)

1 số mà mũ bao nhiêu lần đi nữa thì được 1 số sẽ chia hết cho số ban đầu

\(Vì\) \(5;5^2;5^3;5^4;5^5;...5^{2016}\) đều chia hết cho 5

Các số hạng trong 1 tổng đều chia hết cho 1 số thì tổng đó chia hết cho số đã cho

\(\Rightarrow\)\(5+5^2+5^3+5^4+...+5^{2016}⋮5\) và là hợp số

Vậy C là hợp số

Số thứ 2 là:

\(68708:2=34354\)

Tổng của 3 số là:

\(43756x3=131268\)

Số thứ 3 là:

\(131268-68708-34354=28206\)

Đáp số:28206

Vì \(\dfrac{1}{11}>\dfrac{1}{18}>\dfrac{1}{21}>\dfrac{1}{24}>\dfrac{1}{27}>\dfrac{1}{29}\)

\(\Rightarrow\dfrac{1}{11}+\dfrac{1}{11}+\dfrac{1}{11}+\dfrac{1}{11}+\dfrac{1}{11}+\dfrac{1}{11}+\dfrac{1}{11}>\dfrac{1}{4}+\dfrac{1}{11}+\dfrac{1}{18}+\dfrac{1}{21}+\dfrac{1}{24}+\dfrac{1}{27}+\dfrac{1}{29}\)\(\dfrac{1}{11}+\dfrac{1}{11}+\dfrac{1}{11}+\dfrac{1}{11}+\dfrac{1}{11}+\dfrac{1}{11}+\dfrac{1}{11}=\dfrac{1}{11}.7=\dfrac{7}{11}\)

Ta có:

\(\dfrac{7}{11}=\dfrac{7.5}{11.5}=\dfrac{35}{55};\dfrac{4}{5}=\dfrac{4.11}{5.11}=\dfrac{44}{55}\)

\(Vì\) \(\dfrac{44}{55}>\dfrac{35}{55}\)

\(\Rightarrow\dfrac{4}{5}>\dfrac{7}{11}\)

\(\Rightarrow\dfrac{1}{4}+\dfrac{1}{11}+\dfrac{1}{18}+\dfrac{1}{21}+\dfrac{1}{24}+\dfrac{1}{27}+\dfrac{1}{29}< \dfrac{4}{5}\left(đpcm\right)\)

\(\left(7x-11\right)^3=2^5.5^2+2.10^2\)

\(\left(7x-11\right)^3=2^5.5^2+2.\left(2.5\right)^2\)

\(\left(7x-11\right)^3=2^5.5^2+2.2^2.5^2\)

\(\left(7x-11\right)^3=2^5.5^2+2^3.5^2\)

\(\left(7x-11\right)^3=5^2\left(2^5+2^3\right)\)

\(\left(7x-11\right)^3=5^2\left(32+8\right)\)

\(\left(7x-11\right)^3=5^2.40\)

\(\left(7x-11\right)^3=25.40\)

\(\left(7x-11\right)^3=1000\)

\(\left(7x-11\right)^3=10^3\)

\(\Rightarrow7x-11=10\)

\(\Rightarrow7x=10+11\)

\(\Rightarrow7x=21\)

\(\Rightarrow x=21:7\)

\(\Rightarrow x=3\)

\(80-7\left(x-7\right)=95-3^2.6\)

\(80-7\left(x-7\right)=95-9.6\)

\(80-7\left(x-7\right)=95-54\)

\(80-7\left(x-7\right)=41\)

\(7\left(x-7\right)=80-41\)

\(7\left(x-7\right)=49\)

\(x-7=49:7\)

\(x-7=7\)

\(x=7+7\)

\(\Rightarrow x=14\)

\(\dfrac{1}{3x4}+\dfrac{1}{4x5}+\dfrac{1}{5x6}+\dfrac{1}{6x7}+\dfrac{1}{7x8}+\dfrac{1}{8x9}+\dfrac{1}{9x10}\)

\(=\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{8}+\dfrac{1}{8}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{10}\)

\(=\dfrac{1}{3}-\dfrac{1}{10}\)

\(=\dfrac{10}{30}-\dfrac{3}{30}\)

\(=\dfrac{7}{30}\)