Question: 1. Find a closed form and a recurrence relation for the sequence 0, 2, 8, 24, 64, 160, 384, 896, 2048, 4608, 10240,... .
2. Find a closed form for the sequence 0, 2, 10, 28, 60, 110, 182, 280, 408, 570, 770,... .
3. Consider the recurrence an = 2an-1 +an-2 -2an-3.
(a) The characteristic equation has three roots, r1, r2, r3. What are they?
(b) Find initial conditions so that an = (r1)n is the closed form for the recurrence. Use the recurrence to generate a few more terms to be sure your initial conditions work.
(c) Find initial conditions so that an = (r1)n +(r2)n is the closed form for the recurrence. Use the recurrence to generate three additional terms to be sure your initial conditions work.
(d) Find initial conditions so that an = (r1)n +(r2)n +(r3)n is the closed form for the recurrence. Use the recurrence to generate three additional terms to be sure your initial conditions work.