3. Show that (1.2)+(2-3)+(3.4) + ... + n(n+1) = n(n+1)(n+2) for all natural numbers n = 1,2,3,... 3 4. Show that n2 + 3n is divisible by 2 for all natural numbers n 2 1
(a) Prove that, for all natural numbers n, 2 + 2 · 2 2 + 3 · 2 3 + ... + n · 2 n = (n − 1)2n+1 + 2. (b) Prove that, for all natural numbers n, 3 + 2 · 3 2 + 3 · 3 3 + ... + n · 3 n = (2n − 1)3n+1 + 3 4 . (c) Prove that, for all natural numbers n, 1 2 + 42 + 72...
demonstrates the validity for all n belonging to N (natural numbers) a) divisible by 3 b) divisible by 9 c) divisible by 13 d) divisible by 64 Demostrar la validez de las siguientes afimaciones para todo n e N. a) 2n (-1)n+1 es divisible entre 3, b) 10 3 4n+1 +5 es divisible entre 9, c) 52n (1)"+1 es divisible entre 13, d) 72n 16n - 1 es divisible entre 64,
2) Show that the set that contains all the subsets of the natural numbers N (i.e. the power set of N usually denoted by 2) is uncountable.
24. a) Show that the product of three consecutive numbers is divisible by 3!. b) Show that the product of r consecutive numbers is divisible by r!. e if 25. Solve for n in n! = 12 x C2
Prove by Induction 24.) Prove that for all natural numbers n 2 5, (n+1)! 2n+3 b.) Prove that for all integers n (Hint: First prove the following lemma: If n E Z, n2 6 then then proceed with your proof.
3. Show that (1-2) + (2.3) + (3.4) + ... + n(n+1) = for all natural numbers n = 1,2,3,... n(n+1)(n+2) 3
*these questions are related to Matlab The number 24 is exactly divisible by eight numbers (i.e. 1, 2, 3, 4, 6, 8, 12 and 24). The number 273 is also exactly divisible by eight numbers (i.e. 1, 3, 7, 13,21, 39, 91 and 273) There are 10 numbers in the range of 1:100 that are exactly divisible by eight numbers (i.e. 24, 30, 40, 42, 54, 56, 66, 70, 78 and 88). How many numbers in the range of n-1:20000...
Problem 6, (20 pts) How many natural numbers n є N between 1 and 100 are there which are not divisible by 5 nor divisible by 7?
Exercise 4.17. a) Find the least integer k such that 2n3 3n2 +3n 1 for all n 2 k 351r 1 A, then 1, being the smallest member of N would also be the smallest A and i $ A for all 1 s i s n, then n +1 would be the smallest member member of A. of A. Exercise 4.10. Show that the sum of the first n odd numbers is equal to n2; that is, show that,...