Prove that the sum of the first n odd positive integers is n2. In other words, show that 1 + 3 + 5 + .... + (2n + 1) = (n + 1)2 for all n ∈ N.
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- Prove or disprove that for any x ∈ N, x(x+1)/2 ∈ N (where N = {0, 1, 2, 3, ….}Prove by Induction that for all integers n ≥ 1, n < n2 + 1 .Yes this problem is silly, but still do it by induction! Prove by Induction that for all integers n ≥ 3, 2n < n2 .Prove that if x∈R and x >−1, then(1 +x)n≥1 +nx for all n∈N
- . Prove that n2 + 1 2n, where n is a positive integer with 1 n 4.Show that for f(n) = 2n2 and g(n) = 20n + 3n2 , f(n) is θ(g(n)). How many ways can it be shown? Also Show that for g(n) = 10n2and f(n) = n! + 3 , f(n) is Ω(g(n)). How many ways can it be shown? Discuss with the instructor.Show by induction that T(n) ≤ 3n for all n ≥ 1
- Prove that they are indeed equal: (n + 1)! - 1 + (n + 1)(n + 1)! = (n + 2)! -1Informal Proofs Use strong induction to show that every positive integer, n, can be written as a sum of distinct powers of two: 20, 21, 22, 23, ...:1 = 20, 2 = 21, 3 = 20 + 21, ....(i) Prove by cases for any given integer n,the number (n3-n) is even. (Ii)Prove by contradiction there are no integers x and y such that x2=4y+2.