Please answer the 8th question. I. (2 points) Let F, be the Fibonacci numbers, w

Question

Please answer the 8th question.

I. (2 points) Let F, be the Fibonacci numbers, where Fo= l, F,-I, F2-2, F,-3, F. = 5 . Prove -2 F1 = FN-2, where N> 2 using induction. (Lipschutz, HW 3) 2 C2 points) Use induction to prove that fr all natural numbersas divisible by x-1. Heileman, p.415) 3. (2 points) Prove by induction that l 2 + 22 + 32 + + n2 4. (2 points) Prove by induction that the sum of the first n odd positive integers is n, i.e., 1 +3+5 (2n)-r 5, (2 points) Prove that -ok2k-(n-1)2n+1 + 2. (Heileman, p. 415) 6. (2 points) Assuming a and b are arbitrary constants, and 0

Explanation / Answer

8) 1) x^2 + 3x -10 is clearly O(n^2) as the degree of input polynomial is 2

2) This can be represented as (x+5)(x-2) so it will be Omega(x) * Omega(x) = Omega(x^2).

Since it is both O(x^2) and Omega(X^2) hence it is actually theta(X^2) as required in the question.

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