\(1+\frac{1}{a^2}+\frac{1}{\left(a+1\right)^2}=\frac{\left\lbrack a\left(a+1\right)\right\rbrack^2+\left(a+1\right)^2+a^2}{a^2\cdot\left(a+1\right)^2}\)
\(=\frac{a^2\left(a^2+2a+1\right)+a^2+2a+1+a^2}{a^2\cdot\left(a+1\right)^2}=\frac{a^4+2a^3+a^2+2a^2+2a+1}{a^2\cdot\left(a+1\right)^2}\)
\(=\frac{a^4+2a^3+3a^2+2a+1}{a^2\cdot\left(a+1\right)^2}=\frac{a^4+a^3+a^2+a^3+a^2+a+a^2+a+1}{a^2\cdot\left(a+1\right)^2}\)
\(=\frac{a^2\left(a^2+a+1\right)+a\left(a^2+a+1\right)+\left(a^2+a+1\right)}{a^2\cdot\left(a+1\right)^2}=\frac{\left(a^2+a+1\right)^2}{\left(a^2+a\right)^2}\)
\(=\left(\frac{a^2+a+1}{a^2+a}\right)^2=\left(1+\frac{1}{a^2+a}\right)^2=\left(1+\frac{1}{a}-\frac{1}{a+1}\right)^2\)
\(A=\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+\cdots+\sqrt{1+\frac{1}{11^2}+\frac{1}{12^2}}\)
\(=\sqrt[2]{\left(1+\frac12-\frac13\right)^2}+\sqrt{\left(1+\frac13-\frac14\right)^2}+\cdots+\sqrt{\left(1+\frac{1}{11}-\frac{1}{12}\right)^2}\)
\(=1+\frac12-\frac13+1+\frac13-\frac14+\cdots+1+\frac{1}{11}-\frac{1}{12}=\left(1+1+\cdots+1\right)+\left(\frac12-\frac{1}{12}\right)\)
\(=10+\frac12-\frac{1}{12}=10+\frac{5}{12}=\frac{125}{12}\)
