$\text{ Let }{t}_r\text{ denotes the rth term in the binomial expansion of }(1+a{)}^{50}\text{ . }$$\text{ If }{t}_{25}+t_{27}=\frac{125}{52}{t}_{26}\text{ Then sum of all possible values of a is }$

We have

$\frac{t_{r+1}}{t_r}=\frac{{ }^{50}{C}_r{a}^r}{{ }^{50}{C}_{r-1}{a}^{r-1}}=\left(\frac{51-r}{r}\right)a$

       Now

$\begin{array}{l}\frac{t_{25}}{t_{26}}+\frac{t_{27}}{t_{26}}=\frac{125}{52}\\ \Rightarrow \frac{26}{25a}+\frac{25a}{26}=\frac{125}{52}\\ \Rightarrow \frac{25}{26}{a}^2-\frac{125}{52}a+\frac{26}{25}=0\\ \text{ Sum of all possible values of }a\text{ is }\\ \frac{125}{52}\times \frac{26}{25}=\frac{5}{2}\end{array}$