8) Let a=7a - 1 - 120,-2,4, =1, a, =2.Use the strong induction to show that...
solve 3.44 by strong induction Statement 3.44. Let a = 1, az = 3, and for each natural number greater than 2 define an = an-1 + an-2. Then an < (7/4)" for all natural numbers n. Statement 3.45. Let aj = 1, a2 = 2, a3 = 3, and define an = an-1 + for all 4 Thong on for all incN
Use strong induction to show that every positive integer can be written as a sum of distinct powers of two (i.e., 20 = 1; 21 = 2; 22 =4; 23 = 8; 24 = 16; :). For example: 19 = 16 + 2 + 1 = 2^4 + 2^1 + 2^0 Hint: For the inductive step, separately consider the case where k +1 is even and where it is odd. When it is even, note that (k + 1)=2 is...
4. (25 Pts) Use the strong form of Induction to prove that for all integers 4 where a1 1, a2 3, an - an-1 + an-2 for n 2 3 an-2 for n23.
8. Use induction (on n) to show that: (a) (2n)! > 2" (n!)?, for n > 0. (w) (%) = (2+1), for os ms m. (0) Ž -»* (*) = (–1"("m"), for os m<n.
1. Consider the following sequence: a,-1+30-4 a,-1+30 +3a,-16 а,-1+3a0 + 3a1 +3a2-64 Use Weak Mathematical Induction (on homework 7A, you used Strong) to prove that an-4 for all n2 0. (a) State and prove the Base Case: (b) State the Inductive Hypothesis (c) Show the Inductive Step
Use the Principle of mathematical induction to prove 2. Use the Principle of Mathematical Induction to prove: Lemma. Let n E N with n > 2, and let al, aa-.., an E Z all be nonzero. If gcd(ai ,aj) = 1 for all i fj, then gcd(aia2an-1,an)1. 1, a2,, an
Instructions: Please show all of your work. Unsupported answers may receive no credit. 1. (20 pts) Use mathematical induction to show that for integers n 21, 2.21 +3.22 + ... + (n + 1)21 = n. 21+1 w 2. (20 pts) Let P(n) be statement that a postage of n cents can be formed using only 4-cent and 7-cent stamps. Using strong induction, prove P(n) is true for n 2 18.
(a) Use mathematical induction to prove that for all integers n > 6, 3" <n! Show all your work. (b) Let S be the subset of the set of ordered pairs of integers defined recursively by: Basis Step: (0,0) ES, Recursive Step: If (a, b) ES, then (a +2,5+3) ES and (a +3,+2) ES. Use structural induction to prove that 5 (a + b), whenever (a, b) E S. Show all your work.
Problem 8: (i) Use the Principle of Mathematical Induction to prove that 2n+1(-1)" + 1 1 – 2 + 22 – 23 + ... + (-1)22" = for all positive integers n. (ii) Use the Principle of Mathematical Induction to prove that np > n2 + 3 for all n > 2.
7.3 Practice Problems Prove each of the following statements using mathematical induction. 1. Show that 2 + 4 +8+ ... +2n = 20+1 -2 for all natural numbers n = 1,2,3,... y lo 2. Show that 12 +22+32 + ... + n2 = n(n+1)(2+1) for all natural numbers n = 1,2,3,...