some have the correct answer

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\f\f\fWe will nd the solution to the following lhcc recurrence: an = 76%,1 7 9%,; for n 2 2 with initial conditions do = 2, a1 = 7. The first step as usual is to nd the characteristic equation by trying a solution of the "geometric" format a" = r\". (We assume also 7' y 0). In this case we get: T" : 76T'kl 7 91"k2. Since we are assuming 7' 5A 0 we can divide by the smallest power of 1', Le, 1"\"2 to get the characteristic equation: 72 : 6r 9. (Notice since our lhcc recurrence was degree 2, the characteristic equation is degree 2.) This characteristic equation has a single root 1'. (We say the root has multiplicity 2). Find 1'. 'r='73' Since the root is repeated, the general theory (Theorem 2 in Section 8.2 of Rosen) tells us that the general solution to our lhcc recurrence looks like: a,\" : 01(7')" + agnfr)\" for suitable constants a1, (12. To nd the values of these constants we have to use the initial conditions (10 = 2, a1 = 7. These yield by using n=0 and n=1 in the formula above: 2 : 01(7')\" W a20(r)0 and 7 : cv1(r)1 ,, a21(r)1 By plugging in your previously found numerical value for 1' and doing some algebra, find (11, 012: on :i 2 i m : Li Note the final solution of the recurrence is: an : 0510')\" + earth)\" where the numbers 7', all- have been found by your work. This gives an explicit numerical formula in terms of n forthe an. Find the solution to the following linear, homogeneous recurrence with constant coefcients: an : 3an,1 25%,2 75%,3 for n, 2 3 with initial conditions at, : 8, a1 : 527 a; : 64. The solution is ofthe form: an I (a + 119M110" + (at - iX-ir)" + 73\" for suitable integer constants a, [9,7, 1', a. Note that the variable 7' in this problem doesn't represent a characteristic value. Find these constants and entertheir values: 'r:l The solution can also be written in piecewise form and purely in terms of real numbers: 811'" W 1:55" for 7; mod 4 : 0 (3ng 855" for n mod 4 : 1 a.\" Z 831'\" 1:55" for 7; mod 4 : 2 041'" W 055" for 7: mod 4 : 3 for suitable real constants cl, (:2, 03, C4, c5. Find these constants as well