Question

16 Rachel and Michael have spent all their time studying math (excellent!) and none of it studying for their history exam (also excellent!). To pass history, they develop an advanced version of Morse Code to communicate during the exam. They've encoded some sequences into coughs, pencil taps, and the occasional interpretive dance. Hoping to get in on their knowledge, you record the following sequence of signals. 1 2 4 8 3' 9' 27' 81' 243 If we call the first term ao, a) Find a recursive expression for this sequence b) Find a closed form expression for an. c) Tyanna must also be in on this scheme, because she does a quick hip-hop chicken dance during the exam. You know the appropriate response is either a sprinkler dance or an upside down Gangnam Style. To decide which, you must determine the fixed point(s) of this sequence: -15 an+1 = 8 an d) Here is a table of values for the recursion in part (c). The top row contains 6 different initial values for the sequence. Below each initial value are the first 4 terms of the sequence it would generate. All entries that require a calculator are supplied for you. Fill in the missing entries. -5 -4 -3 -1 ao ai a2 a3 a4 -6 -5.5 -5.2727 -5.1551 -5.0903 -4.25 -4.4705 -4.6447 -4.7705 -10.142 -8.6818 -6.5211 -6.2722 -5.6997 e) Based on the table above, determine whether each of the fixed points you found in part (c) is stable or unstable.

Answer 1

343 2 / au-l an= a> fint recursive exprellion for this sequence solo given sego! / 2 1 2 1 2 1 2 11 16 the recursine relation is An-l, n=1,2,3,.. ewhere do= 1/3 6> closet form expression for an solo ana =) an= 2 X 2 An-2) = 2 X 2 X 2 anos) = 2 x 2 x 2 x... ntimes X 2 X do = (endo = (²9) z) an= 1/3 (22), which is required closed Joom en bussion of an Solo let anet the sequence fired point above sequence then we have **be of x*--15' - 8 X* =) (X*+8X* +15= 0

8 =-15 ao let a = -2 =x* =-5,-3 So fined points are us and -3 to Fill in the milling value. solo - let, ao=-5 =) ai= =-15-8 +3-8=-5 eimilarly we habe az= azz aq = -5 = 0,= - 15 -8 a = -15-8=-15-8=-3 ao -3 azz az=0&=-3 =) 9,= -15 -8 =-0.5 let ao=-3 - 2 =) A₂² 2-15-8=30-8=22 -0.5

-15 let do=-1 9,2-15 -8 z 15-827 So the Complete table is dol-6 -4 1-31-2 1-1 -5.5 -4.25 1-3-0.517 A2 -5.2727-5 -4.4701-322 -10.14 A31-5.15511-5 -4.46 1-3.1-8.681-6.52 94-5.0931-5 1-1.77 1-3 1-6.271-5.69 ai