3 For each positive integer n, define E(n) 2+4++2n (a) Give a recursive definition for E(n)....
Let S(n) be a statement parameterized by a positive integer n. Consider a proof that uses strong induction to prove that for all n 4.S(n) is true. The base case proves that S(4), S(5), S(6), S(7), and S(8) are all true. Select the correct expressions to complete the statement of what is assumed and proven in the inductive step. Supposed that for k> (1?),s() is true for everyj in the range 4 through k. Then we will show that (22)...
(3) Uee mathematical induction to prove that the statement Vne ZtXR<n) → (2n+/< 2")) is true. (Suggestion : Let Ple) dernote the sentence "(2<n)-> (21+k< 20)". In carrying out the proof of the inductive step Van Zyl onafhan) consider the cases PQ)=P(2), P2)->P(3), and Pn>Plitr) for 173, Separately.)
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)
4. Let P(n) be the statement that a postage of n cents can be formed using just 4-cent stamps and 7-cent stamps. The parts of this exercise outline a strong induction proof that P (n) is true forn > 18. a) Show statements P(18), P(19), P (20), and P(21) are true, completing the basis step of the proof. b) What is the inductive hypothesis of the proof? c) What do you need to prove in the inductive step? d) Complete...
50. What is wrong with this "proof? "Theorem For every positive integer n = (n + /2. Basis Step: The formula is true for n = 1. Inductive Step: Suppose that +Y/2. Then -(+972 +*+- +*+1)/2 + + + /- + 1). By the inductive hypothesis, we have + /2-[(++P/2, completing the + inductive step.
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...
3. Let P(n) be the statement that a postage of n cents can be formed using just 3-cent stamps and 5-cent stamps. The 5 / Induction and Recursion parts of this exercise outline a strong induction proof that P(n) is true for n 18. a) Show statements P(18), P(19), P(20), and P(21) are true, completing the basis step of the proof. b) What is the inductive hypothesis of the proof? c) What do you need to prove in the inductive...
Can someone answer number 4 for me? (60 pt., 12 pt. each) Prove each of the following statements using induction. For each statement, answer the following questions. a. (2 pt.) Complete the basis step of the proof b. (2 pt.) What is the inductive hypothesis? c. (2 pt.) What do you need to show in the inductive step of the proof? d. (6 pt.) Complete the inductive step of the proof. 1. Prove that Σ(-1). 2"+1-2-1) for any nonnegative integer...
In the following problem, we will work through a proof of an important theorem of arithmetic. Your job will be to read the proof carefully and answer some questions about the argument. Theorem (The Division Algorithm). For any integer n ≥ 0, and for any positive integer m, there exist integers d and r such that n = dm + r and 0 ≤ r < m. Proof: (By strong induction on the variable n.) Let m be an arbitrary...
The symbol N denotes the nonnegative integers, that is, N= {0,1,2,3,...}. The symbol R denotes the real numbers. In each of the proofs by induction in problems (2), (3), and (4), you must explicitly state and label the goal, the predicate P(n), the base case(s), the proof of the base case(s), the statement of the inductive step, and its proof. Your proofs should have English sentences connecting and justifying the formulas. As an example of the specified format, consider the...