/*! 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 18. In Problem, use the Principle of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Q 18. (page 830)

In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.

1路2+3路4+5.6+...+(2n-1)(2n)=13n(n+1)((4n-1)

Short Answer

Expert verified

The statement is shown.

Step by step solution

01

Step 1. Given information

The statement is1路2+3路4+5.6+...+(2n-1)(2n)=13n(n+1)((4n-1).

02

Step 2. Show the given statement.

Put n=1in the statement 1路2+3路4+5.6+...+(2n-1)(2n)=13n(n+1)((4n-1)

width="177">1.2=13,1.(1+1)(4.1-1)

So, the statement is true for n=1.

Next, we assume that the above statement is true for some k. so that

width="359">1.2+3.4+5.6+鈰�+(2k-1)(2k)=13k(k+1)(4k-1)

Now we consider the sum of first k+1terms.

1.2+3.4+5.6+鈰�+(2k-1)(2k)+(2(k+1)-1)(2(k+1))=13k(k+1)(4k-1)+(2k+2-1)(2(k+1))=(k+1)3(k(4k+3)+2(4k+3))=(k+1)34k2+3k+8k+6=13(k+1)((k+1)+1)(4(k+1)-1)

As a result, statement is true 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 17鈥�28, write down the first five terms of each sequence.

sn=n

Growth of a Rabbit Colony:

A colony of rabbits begins with one pair of mature rabbits, which will produce a pair of offspring (one male, one female) each month. Assume that all rabbits mature in 1 month and produce a pair of offspring (one male, one female) after 2 months. If no rabbits ever die, how many pairs of mature rabbits are there after 7 months?

[Hint: The Fibonacci sequence models this colony. Do you see why?]

In a(n) ______sequence the ratio of successive terms is a constant.

In Problems 71-82, find the sum of each sequence.

鈭�(-k=124k)

In Problems 51鈥�66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.

8+12+18+27+....
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.