3. Let P(n) be the statement that 12 +2²+...+ n² = n(n+ 1)(2n + 1)/6 for the positive integer n. a) What is the statement P(1)? b) Show that P(1) is true, completing the basis step of a proof that P(n) is true for all positive integers n. c) What is the inductive hypothesis of a proof that P(n) is true for all positive integers n? d) What do you need to prove in the inductive step of a proof that P(n) is true for all positive integers n? e) Complete the inductive step of a proof that P(n) is true for all positive integers n, identifying where you use the inductive hypothesis.

3. Let P(n) be the statement that 12 +2²+...+ n² = n(n+ 1)(2n + 1)/6 for the positive integer n. a) What is the statement P(1)? b) Show that P(1) is true, completing the basis step of a proof that P(n) is true for all positive integers n. c) What is the inductive hypothesis of a proof that P(n) is true for all positive integers n? d) What do you need to prove in the inductive step of a proof that P(n) is true for all positive integers n? e) Complete the inductive step of a proof that P(n) is true for all positive integers n, identifying where you use the inductive hypothesis.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

See similar textbooks

Similar questions

Order 7 of the following sentences so that they form a direct proof of the statement: If n is even, then n² + 3n+ 5 is odd. Direct proof of the statement (in order): Since k is an integer, Choose from this list of sentences By the definition of odd, n = 2k + 1 for some integer k. By the definition of even, n = 2k for some integer k. It follows that n² + 3n+ 5 = (2k)² + 3(2k) + 5 = 4k² + 6k+5 = 2(2k² + 3k + 2) + 1 Hence, n² + 3n+ 5 is odd. Let n be odd. Thus, by the definition of odd, . 2(2k² + 3k + 2) + 1 is odd. 2k² + 3k + 2 is even. 2k² + 3k + 2 is an integer. Let n be even.
Define P(n) to be the assertion that: n Σj³ j=1 What does P(3) evaluate to 36 = 36 36 = 14 True 14 = 36 n(n+1) (2n+1) 6
Fill in the words to form a complete proof. Use the given equations in a complete proof of each theorem. Your proof should be expressed in complete English sentences.
can you help explain the question on the attached?
Write a formal proof for the following biconditional statement.For all integers n, n^3 is even if and only if n is even.[Hint: For one of the two directions of this proof, you might want to consider a proof using the contrapositive.][Reminder: You may use the parity property, which states that each integeris either even or odd, but not both.]
Prove or disprove each of the following: [10] 1. (n+a) = 0(nb), for any real constants, where b > 0. 2. 3an = 0(3")
Give all answers
Order the following sentences so that they form a direct proof of the statement: If x and y are rational numbers, then 3x + 2y is also a rational number. Choose from this list of sentences Since b, d are integers, bd is an integer. By the definition of rational numbers, x = a/b where a and b are integers and b# 0, and y = c/d where c and d are integers and d 0. Since 3x + 2y can be expressed as a ratio of two integers where the denominator is not zero, 3x + 2y is a rational number. Since a, b, c, d are integers, 3ad + 2bc is an integer. Let x and y be rational numbers. Since b 0 and d # 0, bd # 0. It follows that 3x + 2y = 3a/b + 2c/d = (3ad + 2bc)/(bd) Direct proof of the statement (in order):
This is Discrete Math. Please show all work and give explanations for solutions. Thank you!