Find the solution to the following lhcc recurrence: an=8anâ1â16anâ2 for nâ¥2 with initial conditions a0=4,a1=7. The first step as usual is to find the characteristic equation by trying a solution of the "geometric" format an=rnan=rn. (We assume also râ 0). In this case we get: rn=8r^nâ1â16r^nâ2. Since we are assuming râ 0râ 0 we can divide by the smallest power of r, i. e., rnâ2 to get the characteristic equation:r^2=8râ16. (Notice since our lhcc recurrence was degree 2, the characteristic equation is degree 2.)This characteristic equation has a single root rr. (We say the root has multiplicity 2). Find r. r=?Since the root is repeated, the general theory (Theorem 2 in Section 8.2 of Rosen) tells us that the general solution to our lhcc recurrence looks like:an=α1(r)n+α2n(r)n for suitable constants α1, α2.To find the values of these constants we have to use the initial conditions a0=4,a1=7. These yield by using n=0 and n=1 in the formula above:4=α1(r)^0+α20(r)^0and7=α1( r)^1+α21(r)^1By plugging in your previously found numerical value for rr and doing some algebra, find α1,,α2:α1=α2=Note the final solution of the recurrence is:an=α1(r)^n+α2n(r)^nwhere the numbers r,αi have been found by your work. This gives an explicit numerical formula in terms of n for the an.

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answer: a video game has three difficulty levels: easy, medium, and expert. the game's scoring works in such a way that a player has to score a fixed number of points to advance to the next level, and then the scoring starts over. the total score needed to clear all three difficulty levels is 7,500. two times the number of points needed to clear the easy level is 500 more than the number of points needed to clear the medium level. twice the number of points needed to clear the medium difficulty level is 1,500 more than the number of points needed to clear the expert level. if x, y, and z represent the easy, medium, and expert levels, respectively, and a is the coefficient matrix of the system of equations modeling this situation, identify the inverse matrix, a-1, and the solution matrix, x, of the system of equations.

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a)(x-2)^2+(y-1)^2=9

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answer: the resulting cross section is a rectangle. solution: a plane parallel to the base of a triangular prism will intersect a cross section that is the same shape as its bases. so the cross section is a triangle.

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