Quiz 2 (c) An even more misguided 6. 270 student designs another self-replicating robot to hunt down and destroy robots of the first kind. The number of these robots at the d of day n is Pn, where P0=0 P1=1 Pn=5Pn-1-6Pn-2+1(for n> 2) Find a closed-form expression for Pn. Show your work clearly to be eligible for rtial credit Solution. The characteristic equation is a-5r+6=0. The right side factors into (a-2)(a-3), so the roots are 2 and 3. For a particular solution, let's first guess Pn =c Substituting this into the recurrence equation gives c 5c-6c+ l, which implies that C=1/ 2. Therefore, the general form of the solution is Pn=A.2+B·3+1/2 Substituting Po=0 and Pi= l gives the equations 0=A+B+1/2 1=2A+3B+1/2 Solving this system gives A=-2 and B=3/2. Therefore, the solution is Pn +1/2Quiz 2 7 (c) An even more misguided 6.270 student designs another self­replicating robot to hunt down and destroy robots of the first kind. The number of these robots at the end of day n is Pn, where: P0 = 0 P1 = 1 Pn = 5Pn−1 − 6Pn−2 + 1 (for n ≥ 2) Find a closed­form expression for Pn. Show your work clearly to be eligible for partial credit. 2 Solution. The characteristic equation is x − 5x + 6 = 0. The right side factors into (x − 2)(x − 3), so the roots are 2 and 3. For a particular solution, let’s first guess Pn = c. Substituting this into the recurrence equation gives c = 5c − 6c + 1, which implies that c = 1/2. Therefore, the general form of the solution is: Pn = A 2n + B 3n · · + 1/2 Substituting P0 = 0 and P1 = 1 gives the equations: 0 = A + B + 1/2 1 = 2A + 3B + 1/2 Solving this system gives A = −2 and B = 3/2. Therefore, the solution is: 3 Pn = −2 · 2n + 3n + 1/2 2 ·