Find a closed form and a recurrence relation for the

Question

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 = 2a_n-1 +a_n-2 -2a_n-3.

(a) The characteristic equation has three roots, r₁, r₂, r₃. What are they?

(b) Find initial conditions so that a_n = (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 a_n = (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 a_n = (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.