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MA/CS 109 DISCUSSION 4 Brian Borucki – [email protected] Problem 4 Gen 1 --------------------- 2 6 2 ------------- 4 14 8 3 ----- How many people are in a given generation? 2, 4, 8, 16, … 𝟐𝒎 (where m is the generation) Problem 4 • So the pattern that emerges is this: M (# of generations) N (# of people in fam) 1 2 2 6 3 14 4 30 … … • Your task is to find two equations: • One that goes from N -> M • Another that goes from M -> N Finding N given M • Given: M (# of generations) N (# of people in fam) N+2 1 2 4 =4 22 2 6 8 =8 23 3 14 1616 = 24 4 30 3232 = 25 … … … • We notice that every N is consistently 2 away from the form: 2𝑖 • We can then notice that the exponent is always one more than the generation • This gives us: N = 2𝑀+1 − 2 Finding M given N • Now that we know: N = 2𝑀+1 − 2, all we need to do is solve for M 𝑁 = 2𝑀+1 − 2 𝑁 + 2 = 2𝑀+1 log 𝑁 + 2 = log 2𝑀+1 log 𝑁 + 2 = 𝑀 + 1 𝑀 = log 𝑁 + 2 − 1 𝑁+2 = log( ) 2 (add 2 to both sides) (take the log of both sides) (simplify) Summing it all up • How do we compare these functions? • By how they grow 10 8 6 N^2 (N^2)/2 N log(N) 4 2 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 -2 Range: [0,3] Summing it all up • How do we compare these functions? • By how they grow 120 100 80 N^2 (N^2)/2 x log(x) 60 40 20 0 1 2 3 4 5 6 7 8 Range: [0,10] 9 10