Evaluate $\frac{1}{m_1} +\frac{1}{n_1} +\frac{1}{m_2} +\frac{1}{n_2}+...+\frac{1}{m_{2011}} +\frac{1}{n_{2011}}$ When $a=1,2,3,...,2010,2011$ the roots of the equation $x^2 -2x-a^2-a=0$ are $(m_1,n_1 ), (m_2,n_2 ), (m_3,n_ 3),..., (m_{2010},n_{2010} ), (m_{2011},n_{2011 }) $ respectively. Evaluate $\frac{1}{m_1} +\frac{1}{n_1} +\frac{1}{m_2} +\frac{1}{n_2}+...+\frac{1}{m_{2010}} +\frac{1}{n_{2010}} +\frac{1}{m_{2011}} +\frac{1}{n_{2011}}$
**My attempt **$$$$
$ \frac{1}{m_a} +\frac{1}{n_a}$can be manipulated to give
$$\frac{m_a+n_a}{m_an_a}$$then w.l.o.g $m_a\ge n_a$
And since$$m_a=2\frac{2+2\sqrt{a^2+a+1}}{2}$$
And $$n_a= 2\frac{2-2\sqrt{a^2+a+1} }{2}$$
We get
$$\frac{m_a+n_a}{m_an_a}=\frac{-2}{a(a+1)}$$
But I dont know what to do next,
Suggestions and solutions would be appreciated
Taken from the 2011 IWYMIC

Hint:
$$
\frac{-2}{a(a+1)} = \frac2{a+1} - \frac 2a
$$
