Explanation Given: Statements says 2 + 4 + 6 + ⋯ + 2 n = n 2 + n + 2 obeys the Condition II of Principle of Mathematical Induction and the statement is false for n = 1 . Formula used: The Principle of Mathematical Induction Suppose that the following two conditions are satisfied with regard to a statement about natural numbers: CONDITION I: The statement is true for the natural number 1. CONDITION II: If the statement is true for some natural number k , it is also true for the next natural number k + 1 . Then the statement is true for all natural numbers. Proof: Consider the statement 2 + 4 + 6 + ⋯ + 2 n = n 2 + n + 2 -----(1) Step 1: Show that statement (1) is true for Condition II. Assume that the statement is true for some natural number k. That is 2 + 4 + 6 + ⋯ + 2 k = k 2 + k + 2 -----(2) Step 2: Prove that the statement is true for the next natural number k + 1 ...

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ISBN: 9780321979070

Author: Michael Sullivan

Publisher: PEARSON

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Chapter 12.4, Problem 29AYU

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To prove: The given statement $2 + 4 + 6 + \dots + 2 n = n^{2} + n + 2$ obeys the Condition II of Principle of Mathematical Induction and the statement is false for $n = 1$ .

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Chapter 12.4, Problem 28AYU

Chapter 12.4, Problem 30AYU

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To prove: The given statement 2 + 4 + 6 + ⋯ + 2 n = n 2 + n + 2 obeys the Condition II of Principle of Mathematical Induction and the statement is false for n = 1 .

Solution Summary: The author explains the principle of mathematical induction and how to prove that the statement is true for all natural numbers.

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To prove: The given statement $2 + 4 + 6 + \dots + 2 n = n^{2} + n + 2$ obeys the Condition II of Principle of Mathematical Induction and the statement is false for $n = 1$ .

Want to see the full answer?

Chapter 12 Solutions

Additional Math Textbook Solutions

To prove: The given statement 2 + 4 + 6 + ⋯ + 2 n = n 2 + n + 2 obeys the Condition II of Principle of Mathematical Induction and the statement is false for n = 1 .

Solution Summary: The author explains the principle of mathematical induction and how to prove that the statement is true for all natural numbers.

Concept explainers

Videos

To prove: The given statement 2 + 4 + 6 + ⋯ + 2n = n2 + n + 2 obeys the Condition II of Principle of Mathematical Induction and the statement is false for n = 1.

Want to see the full answer?

Chapter 12 Solutions

Additional Math Textbook Solutions

To prove: The given statement $2 + 4 + 6 + \dots + 2 n = n^{2} + n + 2$ obeys the Condition II of Principle of Mathematical Induction and the statement is false for $n = 1$ .