Write the Butcher table of the Runge-Kutta method defined by $$y_{n+1} = y_n + \frac{h}{2} f(x_n, y_n) + \frac{h}{2} f(x_n + h, y_{n+1})$$ and determine its order of convergence.
The answer by Lutz is really helpful. Thanks. Now
$$b_1=1/2, b_2=1/2, c_1=0, c_2=1, a_{11}=0, a_{12}=0, a_{21}=1/2, a_{22}=1/2$$
Can I just use $\sum_{i=1}^{2}=1$ and $\sum_{i=1}{2}b_ic_i=1/2$ to prove the consistency order is $0$?
Give the function values their traditional names $$ k_1=f(x_n,y_n),~~k_2=f(x_{n+1},y_{n+1}) $$ then $$ y_{n+1}=y_n+\frac{h}2(k_1+k_2) $$ and consequently also $$ k_2=f(x_n+h,y_n+h(\tfrac12k_1+\tfrac12k_2)). $$