Xét vế trái của biểu thức (VT)
\(VT=1+2^2\left(1+1\right)+3^2\left(2+1\right)+4^2\left(3+1\right)+...+10^2\left(9+1\right)\)
\(VT=1+1.2^2+2^2+2.3^2+3^2+3.4^2+4^2+...+9.10^2+10^2\)
\(VT=\left(1+2^2+3^2+4^2+...+10^2\right)+\left(1.2^2+2.3^2+3.4^2+...+9.10^2\right)\)
Đặt \(A=1+2^2+3^2+4^2+...+10^2\)
\(A=1+2\left(1+1\right)+3\left(2+1\right)+4\left(3+1\right)+...+10\left(9+1\right)\)
\(A=1+1.2+2+2.3+3+3.4+4+...+9.10+10\)
\(A=\left(1+2+3+4+...+10\right)+\left(1.2+2.3+3.4+...+9.10\right)\)
Đặt \(B=1.2^2+2.3^2+3.4^2+...+9.10^2\)
\(B=1.2\left(3-1\right)+2.3\left(4-1\right)+3.4\left(5-1\right)+...+9.10\left(11-1\right)\)
\(B=1.2.3-1.2+2.3.4-2.3+3.4.5-3.4+...+9.10.11-9.10\)
\(B=\left(1.2.3+2.3.4+3.4.5+...+9.10.11\right)-\left(1.2+2.3+3.4+...+9.10\right)\)
\(VT=A+B=\left(1+2+3+4+...+10\right)+\left(1.2.3+2.3.4+3.4.5+...+9.10.11\right)\)
Đặt \(C=1+2+3+4+...+10=55\) (Tổng cấp số cộng)
Đặt \(D=1.2.3+2.3.4+3.4.5+...+9.10.11\)
\(4D=1.2.3.4+2.3.4.4+3.4.5.4+...+9.10.11.4\)
\(4D=1.2.3.4+2.3.4\left(5-1\right)+3.4.5\left(6-2\right)+...+9.10.11\left(12-8\right)\)
\(4D=1.2.3.4-1.2.3.4+2.3.4.5-2.3.4.5+3.4.5.6-...-8.9.10.11+9.10.11.12\)
\(4D=9.10.11.12\Rightarrow D=3.9.10.11\)
\(\Rightarrow VT=C+D=55+3.9.10.11=3025=55^2\)
\(\Rightarrow55^2=\left(x+1\right)^2\Rightarrow55=x+1\Rightarrow x=54\)
