(Solved) : Exercise 11 Prove Sum First N Odd Integers N2 1 3 5 2n 1 N2 Exercise 12 Prove N Q29933586 . . .
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Exercise 11. Prove that the sum of the first n odd integers is n2. That is 1+3+5+.. (2n 1) n2 Exercise 12. Prove that n<2n Exercise 13. Prove that n3n for all positive integers. That is Vn EN, n3 n mod 3 Exercise 14. Let f(n) = 2n + 1, n = 0.1.2 Define f(n) recursively Exercise 15. Let f(n)n1,2. Define f(n) recursively Exercise 16. Prove that log2(n!) 2 n, for all n 24. Exercise 17. Prove that 133 divides 11+122-1, for all n 21 2″ Exercise 18. For n E { 1, 2, 3, ), define f(n) = n. Show that ? f(i)2 1 + n 2 Exercise 19. Using the definition of f(n) in Exercise 16, show that > f(i) 1 + n Exercise 20. In this exercise we show, using strong induction, that when n > 2, fn > where fnn-1+n-2 with fo = 0 and fi = 1 a) Let ?-1+ b) Show that c) Now assume that fk-Sk-2 for all k E {3,4, Show that ? is a root for the equation (z + 1) , n), and show that fn+1 > ??+1-2. Hint use the result of a) FYI: n and are known by the Fibonacci sequence and the Golden Ratio. Show transcribed image text
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Answer to Exercise 11 Prove Sum First N Odd Integers N2 1 3 5 2n 1 N2 Exercise 12 Prove N Q29933586 . . .
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