Prove and explain how to represent 2n - 1 in binary notation
2^n - 1 if n is 1, then 2^1 - 1 = 2 - 1 = 1. it's binary form is 1 if n is 2, then 2^2 - 1 = 4 - 1 = 3. it's binary form is 11 if n is 3, then 2^3 - 1 = 8 - 1 = 7. it's binary form is 111 so, 2^n - 1 in binary form is 111...1 => (n 1's)
1. Prove that 1.3....2n-1 1. Prove that-.-. ...--ㄑㄧ for any n E N 2n V2n+1
Please note n's are superscripted. (a) Use mathematical induction to prove that 2n+1 + 3n+1 ≤ 2 · 4n for all integers n ≥ 3. (b) Let f(n) = 2n+1 + 3n+1 and g(n) = 4n. Using the inequality from part (a) prove that f(n) = O(g(n)). You need to give a rigorous proof derived directly from the definition of O-notation, without using any theorems from class. (First, give a complete statement of the definition. Next, show how f(n) =...
(4) Guess a formula for the sum (2n 1) (2n +1) 1.3 3.5 Prove your guess using induction (4) Guess a formula for the sum (2n 1) (2n +1) 1.3 3.5 Prove your guess using induction
Prove: without using l'hopital's rule. infinity 2n-1 ln(2) (2n-1) n infinity 2n-1 ln(2) (2n-1) n
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prove by mathematical induction Prove Ś m2 n(n+1)(2n+1)
Prove that P2n(0)= (-1)n ((2n-1)!!/(2n)!!) using the generation function and a binomial expansion. Show that (sqrt(pi)(4n-1)/(2gamma(n+1)gamma(3/2-n))=(-1)n-1((2n-3)!!/(2n-2)!!)(4n-1)/2n
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Let G = {1, 3, 5, 9, 11, 13} and let represent the binary operation of multiplication modulo 14. (a) Prove that (G, ) is a group. (You may assume that multiplication is associative.) (b) List the cyclic subgroups of (G, ). (c) Explain why (G, ) is not isomorphic to the symmetric group S3. (d) State an isomorphism between (G, ) and (Z6, +).
2n 3. Prove that lim n+on+ 1 2.