/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q 29. Show that the formula2+4+6+...+2... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Q 29. (page 830)

Show that the formula

2+4+6+...+2n=n2+n+2

obeys Condition II of the Principle of Mathematical Induction. That is, show that if the formula is true for some k it is also true for k + 1. Then show that the formula is false for n = 1 (or for any other choice of n).

Short Answer

Expert verified

The given statement is shown.

Step by step solution

01

Step 1. Given information

The given formula is2+4+6+...+2n=n2+n+2.

02

Step 2. Prove the given statement.

Assume that the statement is true for n=k

Thus, 2+4+6+...+2k=k2+k+2

Find the sum for k+1terms and check whether the sum equals (k+1)2+(k+1)+2.

2+4+6+鈥�鈥�鈥�鈥�+2k+2(k+1)=k2+k+2+2k+2=k2+3k+4=k2+2k+1+k+1+2=k2+2k+1+(k+1)+2=(k+1)2+(k+1)+2

Thus, the statement is true for k+1.

The first term in the given series is 2 .

Put n=1in n2+n+2to check if the statement is true for n=1

12+1+2=2+2=4

Thus, the statement is not true forn=1

Since both the conditions of the Principle of Mathematical Induction are not satisfied, the statement is false for all natural numbers.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Problems 61鈥�70, express each sum using summation notation.

1+3+5+7+....+212-1

Environmental Control: The Environmental Protection Agency (EPA) determines that Maple Lake has 250 tons of pollutants as a result of industrial waste and that 10% of the pollutant present is neutralized by solar oxidation every year. The EPA imposes new pollution control laws that result in 15 tons of new pollutant entering the lake each year. The amount of pollutant in the lake at the end of each year is given by the recursively defined sequence

p0=250;pn=0.9pn-1+15

(a) Determine the amount of pollutant in the lake at the

end of the second year. That is, determine p2 .

(b) Using a graphing utility, provide pollutant amounts for

the next 20 years.

(c) What is the equilibrium level of pollution in Maple

Lake? That is, what is localid="1646823556776" limn鈫�鈭�pn?

In Problems 11鈥�16, evaluate each factorial expression.

12!10!

In Problems 17鈥�28, write down the first five terms of each sequence.

an=3nn

In Problems 17鈥�28, write down the first five terms of each sequence.

Sn=2n3n+1
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.