Did you know plushies triple in number each day? It's true!2 Scientists placed a plushie in a single room. Every day, the number of plushies tripled! However, after the third day, the scientists noticed that the plushie from the first day vanished and the remainder had tripled. The next day, the plushies that appeared on the second day vanished as well and the remainder had tripled again. The scientists came to us Discrete experts to ask for help predicting how many new plushies will be appear in the room at any given day. Your task is to do the following: i. Write out the first 5 terms of the sequence given. ii. Define a sequence that represents the number of plushies that appear on day k, assuming the first day is when the scientists place the 1 plushie in the room. In particular, represent the formula for the kth term recursively. That is, your formula for ak must be in terms of previous terms of the sequence like ak1, ak2. For the terms that do not fit your recursive formula, you may simply list them out. For example, if plushies just appeared using the Fibonacci sequence, we would define the sequence as follows ak = ak1 + ak2 for k 3 a2 = 1 a1 = 1 iii. As work for producing the formula, use strong induction to prove your formula accurately depicts the situation after k days. That is, prove the following P (n) holds for n 1 P (n) = If the plushies appear and disappear as observed, then an accurately represents the number of plushies that appear on day n. (Note: You may not need the power of strong induction to prove this claim, but it is available just in case.)