A sequence is defined by Un+1=Un+2Un-1
a) Given that U11=683 and U8=85 deduce the equations

Please clarify, as your notation is confusing. Is this what you mean:


U_{n+1} = U_n + 2U_{n-1}\\
U_{11} = 683, \ U_8 = 85
?

Yes that's exactly what I meant

You can replace U11 =683 with terms consisting of U10 and U9. Then replace the U10 with terms consisting of U9 and U8. Continue until you get U11 solely in terms of U6 and U7. Then do the same for U8=85, getting it in terms of U6 and U7. Now you have two equations in two unknowns (U6 and U7). For this you can calculate the value of Un for any n.