I am trying to understand an example in my book and something isn't making sense to me. The problem is: Use the improved Euler method to calculate approximate values of the solution of the initial value problem $\displaystyle y'=1-t-4y, y(0)=1$.

I have found that $\displaystyle F_0=5$. Assuming that h=.025 we then get $\displaystyle f(t_0+h,y_0+hf_0)=5.475$.

It is after this where I'm getting confused. The equation in my book is$\displaystyle y_n+1=y_n+(h/2)(f_n+f(t_n+h, y_n+hf_n))$. Using this we get $\displaystyle y_1=1+(0.5)(5+5.475)(0.025)=1.1309375$.

I cannot for the life of me figure out where the 0.5 is coming from. Can someone show me where it's coming from?