13)
\(a,A=\sqrt{1+\dfrac{1}{a^2}+\dfrac{1}{\left(a+1\right)^2}}\left(a>0\right)\)
\(=\sqrt{\dfrac{a^2\left(a+1\right)^2+\left(a+1\right)^2+a^2}{a^2\left(a+1\right)^2}}\)
\(=\sqrt{\dfrac{a^4+2a^3+a^2+\left(a+1\right)^2+a^2}{a^2\left(a+1\right)^2}}\)
\(=\sqrt{\dfrac{a^4+2a^2\left(a+1\right)+\left(a+1\right)^2}{a^2\left(a+1\right)^2}}\)
\(=\sqrt{\dfrac{\left(a^2+a+1\right)^2}{a^2\left(a+1\right)^2}}=\left|\dfrac{\left(a^2+a+1\right)}{a\left(a+1\right)}\right|=\dfrac{a^2+a+1}{a^2+a}=1+\dfrac{1}{a\left(a+1\right)}\)
\(b,P=\sqrt{1+\dfrac{1}{1^2}+\dfrac{1}{2^2}}+\sqrt{1+\dfrac{1}{2^2}+\dfrac{1}{3^2}}+...+\sqrt{1+\dfrac{1}{99^2}+\dfrac{1}{100^2}}\)
\(=1+\dfrac{1}{1.2}+1+\dfrac{1}{2.3}+...+1+\dfrac{1}{99.100}\)
\(=1.99+\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{99}-\dfrac{1}{100}\right)\)
\(=99+1-\dfrac{1}{100}=100-\dfrac{1}{100}=\dfrac{9999}{100}=99,99\)
