I'm self studying this book "Methods of Solving Number theory Problems by Elina" since many days but currently stuck on this formula of continued fractions. 
For example $a=87/ 55 = [1,1,1,2,1,1,4]$ in continued fraction. but when I put the values according to the formula I don't get the answer to being $a=87/55$. what am I missing, if someone can explain this in simple words, it will be very helpful. Thanks
We have
$$\frac{P_0}{Q_0}=\frac{1}{1},\frac{P_1}{Q_1}=\frac21, \frac{P_2}{Q_2}=\frac32, \frac{P_3}{Q_3}=\frac85, \frac{P_4}{Q_4}=\frac{11}{7}$$
and
$$\alpha_0 = \frac{87}{55}, \alpha_1 = \frac{55}{32}, \alpha_2 = \frac{32}{23}, \alpha_3 = \frac{23}{9}, \alpha_4 = \frac95, \alpha_5 = \frac54.$$
In particular, we have $\alpha_5 = \frac{5}{4}, P_4 =11, P_3=8, ,Q_4=7, Q_3 = 5 $.
We can verify that $$\frac{P_4\cdot \alpha_5+P_3}{Q_4\cdot \alpha_5 + Q_3}=\frac{11 \cdot \frac54 + 8}{7 \cdot \frac54 + 5}=\frac{55+32}{35+20}=\frac{87}{55}=\alpha$$
Another example, we have $\alpha_4=\frac95, P_3=8, P_2=3, Q_3=5, Q_2=2.$
We can verify that
$$\frac{P_3\cdot \alpha_4+P_2}{Q_3\cdot \alpha_4 + Q_2}=\frac{8 \cdot \frac95 + 3}{5 \cdot \frac95 + 2}=\frac{72+15}{45+10}=\frac{87}{55}=\alpha$$