I have to use $20$ steps of size $h=0.5$ to estimate $y(11)$ for the curve of $y' = x-y$, going through $y(1)=1$.
Now, I could just solve the differential equation, but there will be harder differential equations in the future.
I write this as a recurrence relation. I let $a_{n+1}$ be the next step's $y$, $a_n$ be the current $y$, and let $1+\frac{n}{2}$ be x, because I am using a step size of $0.5$, and my first $x$ is $1$.
So, I should have $y_2=hy'+y$.
Plugging in my variables, this gives me $a_{n+1}=0.5(1+\frac{n}{2}-a_n)+a_n.$
Now, I simplify to get $a_{n+1}=0.5+\frac{n}{4}+\frac{a_n}{2}=\frac{1}{4}(2a_n+n+2).$
So finally, I can punch $a_{n+1}=\frac{1}{4}(2a_n+n+2)$ into my calculator and pray that it generates a table or something.
However, I think I can go farther. I think I could solve the recurrence relation.
How would I go about solving this?