In this case, system (1.8) is expressed as (af (Xn-1)+bf (*n-2)), Yn+1 = f (af (yn-1)+bf (yn-2)), (2.1) Xn+1 = for n e No. Since f is "1 – 1", from (2.1) f (Xn+1) = af (Xn-1)+bf (x,-2), f(Vn+1) = af (yn-1)+bf (yn-2), (2.2) for n E No. By using the change of variables f (xn) = un, and f (yn) = Vn, n>-2, (2.3) system (2.2) is transformed to the following one Un+1 = aun-1 + bun-2, Vn+1 = avn-1+ bvn-2, (2.4) for n e No. By taking a = 0, b= a, c= b in (1.4) and S, = Jn+1, for all n > -2, which is called generalized Padovan sequence, in (1.5), the solutions to equations in (2.4) are given by Un = uoJn+1+u-1Jn+2+ bu_2Jn, (2.5) Vn = VOJN+1+v-1Jn+2+bv_2Jn; (2.6) for n E No. From (2.3), (2.5) and (2.6), it follows that the general solution to system (2.2) is given by Xn = f(f (x0) Jn+1+f (x-1)Jn+2+bf (x-2) Jn), n> –2, Yn = f (f (vo) Jn+1+f (y-1)Jn+2+bf (y-2) Jn), n> -2. (2.7) (2.8) 2.2. Case 2: Pn = Xn, qn = Xn, I'n = Xn, Sn = Xn In this case, system (1.8) becomes Xn+1 =f(af (Xn-1)+bf (xn-2)), Yn+1 =f(af (xn-1)+bf (xn-2)), (2.9) for n E No. It should be first note that from the equations in (2.9) it immediately follows that x, = yn, for all n EN. From (2.7), the general solution to system (2.9) is Xn = Yn = f(f (xo) Jn+1+f (x=1) Jn+2+bf (x_2)Jn), n € N. (2.10)

Where are the follow up steps (step 4-7)?

Arrange these steps in the correct order to prove that if you have an 8-gallon jug of water and two empty jugs with capacities of 5 gallons and 3 gallons, respectively, then you can measure 4 gallons by successively pouring some of or all of the water in a jug into another jug in the correct order. Rank the options below. Fill the 3-gallon jug from the 5-gallon jug, leaving (3, 2, 3). Empty the 5-gallon jug's contents into the 3-gallon jug, leaving (6, 0, 2), and then fill the 5-gallon jug from the 8-gallon jug, producing (1, 5, 2). Fill the 5-gallon jug from the 8-gallon jug, leaving the contents (3, 5, 0), where the ordered triple is used to record the amount of water in the 8-gallon jug, the 5-gallon jug, and the 3-gallon jug, respectively. Top off the 3-gallon jug from the 5-gallon jug, and there will be (1, 4, 3), with four gallons in the 5-gallon jug. Pour the contents of the 3-gallon jug back into the 8-gallon jug, leaving (6, 2, 0). ❤ ← ←

Translate the point 8 units right and 5 units down