Problem

Prove the following by using the principle of mat…

06:40

Problem 18 Easy Difficulty

Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.
$$
1+2+3+\ldots+n<\frac{1}{8}(2 n+1)^{2} .
$$

Answer

Related Courses

Precalculus

NCERT Class 11 - Math

Chapter 4

Principle of Mathematical Induction

Section 1

Introduction

Discussion

You must be signed in to discuss.

Top Precalculus Educators

Watch More Solved Questions in Chapter 4

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Video Transcript

in this problem of mathematical induction we have approved given a statement using principle of mathematical induction for all end belongs to natural number first. Well given a statement sp orphan And we have given a statement one plus two plus three plus up to plus and Less than one upon 8 who and plus one holy square first we prove forward. We of one year of one is basic statement and one is the smallest national number. So we'll take alleges, you know putting anywhere to one. We have very equal to one. Now we take our it is putting an equal to one here. So we have valuable to two plus 13 square upon eggs. So it will make well too nine Upon a. This is required to greater than one so we can say. And the case they satisfy dissatisfied are too for the we are of one for the given statement. Now we consider P. Of gay or to get them. So we write the given a statement one plus two plus three. Plus up to Get them. And this is less than one upon 8 Who gave plus one hole is work. Now we need the proof. We are of K plus one up. Ok plus one time. So we write LHs considering P R. K. Yeah to get them plus Hey Plus 1 to Talk. Now we could be of gay which is one upon it. To K plus one hole is work. So we write it. Mhm. One upon a. Who? K-us one. Holy square plus K plus one. Now we simplify the expression. Taking L. C. M. So L. C. M will be eight and expand these two K plus one. We have value four case for plus one plus 40 and multiplying this aid in two K plus one. We have eight K plus eight. Now we simplify the numerator. Mhm So it will be 4K square plus 12 G plus nine. I'm on a mm. We can write it as Okay Plus three. Hold you as to gays were four cases were plus to multiply by okay Multiply by three plus threes work. And this is also where two cases where so we can write it. Mhm. Cook a plus three. Holy square upon a. Mhm. Yeah. Again arranging this term. So variety too. K plus one plus one. Holy square upon a. Now we take our bridges are just putting N equal to K plus one. So I saw you are riches of the given statement one upon it too. And plus one hole is square. You should be greater than a plus one. Mhm. Two N plus one. Always were upon it putting in equal to K plus one. So we have to. Okay plus one. Mhm Plus one Whole is good upon eight. These two statements alleges our ages are equal so we can say mhm. Mhm. Given a statement. Yeah, yeah, it's true for all and belongs to national number. And this will be our final answer delivery.

Get More Help with this Textbook