We will find the solution to the following Ihcc recurrence: an=-2an-1+8an-2 for n 2 with initial conditions ao = 1, a = 7. The first step in any problem like this is to find the characteristic equation by trying a solution of the "geometric" format an = r". (We assume also r # 0). In this case we get: Since we are assuming r #0 we can divide by the smallest power of r, i.e., 2 to get the characteristic equation: (Notice since our Ihcc recurrence was degree 2, the characteristic equation is degree 2.) Find the two roots of the characteristic equation r and r. When entering your answers use r r T1= and , T2 = Since the roots are distinct, the general theory (Theorem 1 in section 5.2 of Rosen) tells us that the general solution to our Ihcc recurrence looks like: By plugging in your previously found numerical values for r and r and doing some algebra, find a, 02: [Be careful to note that (-x)" -(x") when n is even, for example (-3)-(3).] = for suitable constants a1, a2 To find the values of these constants we have to use the initial conditions ao = 1, a = 7. These yield by using n=0 and n=1 in the formula above: 2 = p2r 1+8pr-2 = Note the final solution of the recurrence is: -2r + 8. where the numbers ri, ai have been found by your work. This gives an explicit numerical formula in terms of n for the an an = a1(ri)" + a2(7)" 1 = a(71) + a(r) 7 = a(ri) + a2(r2) an = a(ri)" + a(7)"