Cái x đầu tiên không phải x² à. Còn cái vế phải là hỗn số hả

Đặt \(A=1\cdot2+2\cdot3+\cdots+99\cdot100\)

\(=1\left(1+1\right)+2\left(2+1\right)+\cdots+99\left(99+1\right)\)

\(=\left(1^2+2^2+\cdots+99^2\right)+\left(1+2+\cdots+99\right)\)

\(=\frac{99\left(99+1\right)\left(2\cdot99+1\right)}{6}+\frac{99\cdot\left(99+1\right)}{2}\)

\(=\frac{99\cdot100\cdot199}{6}+\frac{99\cdot100}{2}=\frac{99\cdot100\cdot199+3\cdot99\cdot100}{6}\)

\(=\frac{99\cdot100\cdot202}{6}=33\cdot100\cdot101=333300\)

Ta có: \(x^2+\left(x^2+1\right)+\left(x^2+2\right)+\cdots+\left(x^2+99\right)\)

\(=\left(x^2+0\right)+\left(x^2+1\right)+\cdots+\left(x^2+99\right)\)

\(=\left(x^2+x^2+\cdots+x^2\right)+\left(0+1+\cdots+99\right)=100x^2+\frac{99\cdot100}{2}=100x^2+99\cdot50=100x^2+4950\)

Ta có: \(\frac{1\cdot2+2\cdot3+\cdots+99\cdot100}{x^2+\left(x^2+1\right)+\left(x^2+2\right)+\cdots+\left(x^2+99\right)}=50\frac{116}{131}\)

=>\(\frac{333300}{100x^2+4950}=50+\frac{116}{131}\)

=>\(\frac{6666}{2x^2+99}=\frac{6666}{131}\)

=>\(2x^2+99=131\)

=>\(2x^2=32\)

=>\(x^2=16\)

=>\(\left[\begin{array}{l}x=4\\ x=-4\end{array}\right.\)

a) \(xy+3x=5y-2\)
\(\Leftrightarrow x\left(y+3\right)=5y-2\)
\(\Leftrightarrow x=\frac{5y-2}{y+3}\)
\(\Leftrightarrow x=\frac{5\left(y+3\right)-17}{y+3}\)
\(\Leftrightarrow x=5-\frac{17}{y+3}\)
Do x nguyên, y nguyên nên y+3 là Ư(17)

Ta có bảng:
 

Vậy (x;y) là (6;-20);(22;-4);(-12;-2);(4;14)  

b) \(\Leftrightarrow\frac{\frac{99.100.101}{3}}{100x^2+\frac{99.100}{2}}=\frac{6666}{131}\Rightarrow x=\pm4\)

cau b hoi tat nhi

Bài 1:

a) Ta có: \(\frac{2^8\cdot4\cdot13+2^7\cdot8\cdot65}{2^9\cdot39}\)

\(=\frac{2^8\cdot4\cdot13+2^8\cdot4\cdot13\cdot5}{2^9\cdot39}\)

\(=\frac{2^{10}\cdot13\left(1+5\right)}{2^9\cdot13\cdot3}=\frac{6}{3}=2\)

b) Đặt \(A=4+2^2+2^3+2^4+...+2^{20}\)

Ta có: \(A=4+2^2+2^3+2^4+...+2^{20}\)

\(\Rightarrow2A=2^3+2^3+2^4+...+2^{21}\)

Ta có: \(2A-A=2^3+2^{21}-2^2-2^2=8+2^{21}-8=2^{21}\)

hay \(A=2^{21}\)

Vậy: \(4+2^2+2^3+2^4+...+2^{20}=2^{21}\)

b) Ta có B=1.2-1+2.3-2+3.4-1+...+99.100-99

A-B=1+2+3+....+100=5050 chia hết cho  50

khó lắm:))