I am studying sequences as part of A-Level Maths and we are doing the recurrence formula type questions. Say it is $u_{n+1} = u_n - u_{n-1}$. And you were asked to find the values of $u_{13}, u_{63}, u_{89}$ .
How would you do this without knowing the numbers in the sequence before?
HINT :
Let $u_1=\color{red}{a},u_2=\color{green}{b}$. Then, we have $$u_3=b-a$$ $$u_4=(b-a)-b=-a$$ $$u_5=-a-(b-a)=-b$$ $$u_6=-b-(-a)=a-b$$ $$u_7=a-b-(-b)=\color{red}{a}$$ $$u_8=a-(a-b)=\color{green}{b}$$