Show that f (n) is O(g(n)) if and only if g(n) is Ω( f (n)).
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Show that f (n) is O(g(n)) if and only if g(n) is Ω( f (n)).
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- Let the sequence (n) be recursively defined by x1 = √2 and Xn+1 = √√2+xn, n≥ 1. Show that (n) converges and evaluate its limit.Prove or disprove that for any x ∈ N, x(x+1)/2 ∈ N (where N = {0, 1, 2, 3, ….}Prove or disprove "Suppose that m and n are integers. If m > n ≥ 0, then gcd (m, n) = gcd (m − n, n)."
- Prove that if n is divisible by neither 2 nor 5, then there is a number m=11....11 (a string of 1s in base 10) which is divisible by n.Prove by induction that fib(0)+fib(1)+…+fib(n) = fib(n+2)-1, fib(0)=0 fib(1)=1It was claimed that:(a, b) ≤ (c, d) ⇔ (a < c) ∨ (a = c ∧ b ≤ d) defines a well-ordering on N x N. Show that this is actually the case.
- Let, a1 = 3, a2 = 4 and for n ≥ 3, an = 2an−1 + an−2 + n2, express an in terms of n.Prove the following: a. [(a mod n) - (b mod n)] mod n = (a - b) mod n b. [(a mod n) * (b mod n)] mod n = (a * b) mod nShow Let f(.) be a computable, strictly monotonic function, that is, f(n+ 1) > f(n) for all n. Show B = {f(n) | n ∈ N} is recursive.