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 ...

Precalculus Plus MyLab Math with eText -- Access Card Package (10th Edition) (Sullivan & Sullivan Precalculus Titles)

10th Edition

ISBN: 9780321978981

Author: Michael Sullivan

Publisher: PEARSON

See similar textbooks

Concept explainers

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

Videos

Question

Chapter 12.4, Problem 29AYU

To determine

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 $n = 1$ .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12.4, Problem 28AYU

Chapter 12.4, Problem 30AYU

Students have asked these similar questions

The parametric equations of the function are given asx=asin²0, y = acos). Calculate [Let: a=anumerical coefficient] dy d²y and dx dx2

A tank contains 200 gal of fresh water. A solution containing 4 lb/gal of soluble lawn fertilizer runs into the tank at the rate of 1 gal/min, and the mixture is pumped out of the tank at the rate of 5 gal/min. Find the maximum amount of fertilizer in the tank and the time required to reach the maximum. Find the time required to reach the maximum amount of fertilizer in the tank. t= min (Type an integer or decimal rounded to the nearest tenth as needed.)

Thumbi Irrigation Scheme in Mzimba district is under threat of flooding. In order to mitigate against the problem, authorities have decided to construct a flood protection bund (Dyke). Figure 1 is a cross section of a 300m long proposed dyke; together with its foundation (key). Survey data for the proposed site of the dyke are presented in Table 1. Table 2 provides swelling and shrinkage factors for the fill material that has been proposed. The dyke dimensions that are given are for a compacted fill. (1) Assume you are in the design office, use both the Simpson Rule and Trapezoidal Rule to compute the total volume of earthworks required. (Assume both the dyke and the key will use the same material). (2) If you are a Contractor, how many days will it take to finish hauling the computed earthworks using 3 tippers of 12m³ each? Make appropriate assumptions. DIKE CROSS SECTION OGL KEY (FOUNDATION) 2m 1m 2m 8m Figure 1: Cross section of Dyke and its foundation 1.5m from highest OGL 0.5m…

Chapter 12 Solutions

Precalculus Plus MyLab Math with eText -- Access Card Package (10th Edition) (Sullivan & Sullivan Precalculus Titles)

Ch. 12.1 - For the function f( x )= x1 x , find f( 2 ) and f(...Ch. 12.1 - True or False A function is a relation between two...Ch. 12.1 - Prob. 3AYU Ch. 12.1 - True or False The notation a 5 represents the...Ch. 12.1 - True or False If is am integer, then Ch. 12.1 - The sequence a 1 =5 , a n =3 a n1 is an example of...Ch. 12.1 - The notation a 1 + a 2 + a 3 ++ a n = k=1 n a k...Ch. 12.1 - Prob. 8AYU Ch. 12.1 - In Problems 11-16, evaluate each factorial...Ch. 12.1 - Prob. 10AYU

Additional Math Textbook Solutions

Find more solutions based on key concepts

Women’s Heights Suppose college women’s heights are approximately Normally distributed with a mean of 65 inches...

Introductory Statistics

The given expression

Pre-Algebra Student Edition

Comparison tests Use the Comparison Test or Limit Comparison Test to determine whether the following series con...

Calculus: Early Transcendentals (2nd Edition)

Fill in each blank so that the resulting statement is true. The quadratic function f(x)=a(xh)2+k,a0, is in ____...

Algebra and Trigonometry (6th Edition)

The null hypothesis, alternative hypothesis, test statistic, P-value and state the conclusion. To test: Whether...

Elementary Statistics

The table by using the given graph of h.

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

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 $2 + 4 + 6 + \dots + 2 n = n^{2} + n + 2$ 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 $n = 1$ .