Solve a0 = 0, a1 = 1 and an = an−1 + an−2 + 2n , for n ≥ 2. Hints: This one involves a lot more algebra than the problems above. Solving the homogeneous problem will involve using the quadratic formula. The characteristic roots turn out to be 1 ± √ 5 2 (but show the work!). For a particular solution, try an = A2 n . You should find A = 4 (but show the work!). Put those two pieces together to write down the general solution an = A( (1 + √ 5 )/2 )^n + B( (1 − √ 5)/ 2 )^n + 4(2^n ) and the determine the values for A and B by using the two initial conditions, a0 = 0 and a1 = 1. The necessary arithmetic will be somewhat complicated, but not impossible.

Hi yes ik I'm not that good at math but wouldn't it be

a n=1∗1 n =1

to plot points   for x just start at the origin (middle) and go to the left however many times it say(if the number is a positive) if the number is a negative then go right however many times it says if it is 0 go nowhere

now keep you pencil there and go up however many times it says on y(if it is positive) if it is negative then go down however many times it says. if it says 0 go nowhere.

you have just plotted a point! now when you finish plotting all of them connect them with a ruler

plot each point as (-3, 7), (0,4), (3,1), (8,-4)   in the parenthesis on the left of the comma is x and the right is y.

.24

step-by-step explanation:

the probability is   .24

the first player had the highest probability of getting a silver marble

the game could be made fair by adding back the marble that was drawn after each draw