Help with both Part A, B, C, D and E. Consider the following proof for the state

Question

Help with both Part A, B, C, D and E.

Consider the following proof for the statement in previous question. Proof We prove the result by mathematical induction. i. n = 2 and x^2_1 + x^2_2 lessthanorequalto x^2_1 + x^2_2 + 2|x_1||x_2| = (|x_1| + |x_2|)^2. ii. Taking squareroot of both sides, we get Squareroot x^2_1 + x^2_2 lessthanorequalto | |x_1| + |x_2| |lessthanorequalto |x_1| + x_2|. Therefore, base case holds. iii. Suppose that the result holds for some integer k with k greaterthanorequalto 2, that is Squareroot x^2_1 + ... + x^2_k lessthanorequalto |x_1| + ... + |x_k|. iv. Consider n = k + 1. Let A^2 = x^2_1 + ... + x^2_k. Then, v. Squareroot x^2_1 + ... + x^2_k + 1 = Squareroot A^2 + x^2_k + 1 lessthanorequalto |A| + |x_k + 1| = Squareroot A^2 + |x_k + 1|. vi. Then we get, Squareroot A^2 + |x_k + 1|lessthanorequalto |x_1| + ... + |x_k| + |x_k + 1|. vii. Hence, Squareroot x^2_1 + ... + x^2_k + 1 lessthanorequalto |x_1| + ... + |x_k + 1| viii. So the result holds for n = k + 1. Therefore, the result holds for all positive integers n by the principle of mathematical induction. Is this proof correct? (A) Yes (B) No Is this a proof by strong induction? (A) Yes (B) No Which of the followings justifies the inequality Squareroot A^2 + x^2_k + 1 lessthanorequalto |A| + |x_k + 1| in step (v.)? (A) Base case. (B) Induction hypothesis. (C) Previous knowledge. (D) Definition of absolute value. Which of the followings justifies the inequality in step (vi.)? (A) Base case. (B) Induction hypothesis. (C) Previous knowledge. (D) Definition of absolute value. Do you find this proof to be complete? (A) Yes, enough detail is provided. (B) No, each step should be justified.

Explanation / Answer

A)THE PROOF IS CORRECT

B) IT IS BY STRONG INDUCTION

C)WE CAN JUSTIFY WITH PREVIOUS KNOWLEDGE i.e., step ii)

d) IT IS BASED ON INDUCTION

E)YES ENOUGH DETAIL IS GIVEN

Related Questions

Navigate

• Browse (All)
• Browse H
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.