
Question 3 You should be able to answer this question after studying Unit 9. (a) Use...
Question 3 10 marks You should be able to answer this question after studying Unit 9. Use mathematical induction to prove that the following statement is true: 7 + 9 + 11 +. + (2n + 5) = n(n + 6) for all n E N. (10)
Question 4 10 marks You should be able to answer this question after studying Unit 9. (a) Use proof by contraposition to prove that the following statement is true for all integers n: If n2 + 2n + 7 is odd, then n is even. [6] (b) Hence prove that the following statement is true for all integers n: na + 2n + 7 is odd if and only if n is even. [4]
Question 2 10 marks You should be able to answer this question after studying Unit 9. (a) State whether the following statement is true or false. If it is true, give a proof; if it is false, show that it is false. х The real function f(x) = is an even function. X + 5 [3] (Recall from Unit 7 that a function f is even if f(-x) = f(x) for all x in the domain of f.) (b) Prove...
DISCRETE MATHEMATIC For question 1, Use mathematical induction to prove the statements are correct for n ∈ Z+(set of positive integers). 1. Prove that for n ≥ 1 1 + 8 + 15 + ... + (7n - 6) = [n(7n - 5)]/2 For question 2, Use a direct proof, proof by contraposition or proof by contradiction. 2. Let m, n ≥ 0 be integers. Prove that if m + n ≥ 59 then (m ≥ 30 or n ≥...
please answer questions #7-13
7. Use a direct proof to show every odd integer is the difference of two squares. [Hint: Find the difference of squares ofk+1 and k where k is a positive integer. Prove or disprove that the products of two irrational numbers is irrational. Use proof by contraposition to show that ifx ty 22 where x and y are real numbers then x 21ory 21 8. 9. 10. Prove that if n is an integer and 3n...
write a formal proof and state witch proof style you
use
1 1 + +...+ 3.4 n-2 6. (5 pts.) a. What is the first n that P(n) is true? P(n): 4.5 n(n+1) 3n+3 b. (20 pts. Use mathematics induction to prove (write a formal proof). For all ne N, where n is greater than or equal to? (the answer form part a) P(n) is true, where 1 1-2 P(n): Be sure to state which of the three types of...
Problem 1 148pts] (1) I 10pts! Let P(n) be the statement that l + 2 + + n n(n + 1) / 2 , for every positive integer n. Answer the following (as part of a proof by (weak) mathematical induction): 1. [2pts] Define the statement P(1) 2. [2pts] Show that P(1 is True, completing the basis step. 3. [4pts] Show that if P(k) is True then P(k+1 is also True for k1, completing the induction step. [2pts] Explain why...
QUESTION 3 Show all your work on mathematical induction proofs Use mathematical induction to prove the formula for every positive integer n
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) =...
Use mathematical induction to prove that the statement is true for every positive integer n. 1'3+ 24 +3'5 +...+() = (n (n+1)(2n+7))/6 a. Define the last term denoted by t) in left hand side equation. (5 pts) b. Define and prove basis step. 3 pts c. Define inductive hypothesis (2 pts) d. Show inductive proof for pik 1) (10 pts)