Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » Advanced Algebra II: Activities and Homework » Homework: Proof by Induction

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • Featured Content display tagshide tags

    This collection is included inLens: Connexions Featured Content
    By: Connexions

    Comments:

    "This is the "main" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing about them in […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Also in these lenses

  • Busbee's Math Materials display tagshide tags

    This collection is included inLens: Busbee's Math Materials Lens
    By: Kenneth Leroy Busbee

    Click the "Busbee's Math Materials" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Homework: Proof by Induction

Module by: Kenny M. Felder. E-mail the author

Summary: An updated version of the Homework: Proof by Induction module.

Exercise 1

Use mathematical induction to prove that 2 + 4 + 6 + 8 ... 2 n = n ( n + 1 ) 2+4+6+8...2n=n(n+1).

  • a. First, show that this formula works when n = 1 n=1.
  • b. Now, show that this formula works for ( n + 1 ) (n+1), assuming that it works for any given n n.

Exercise 2

Use mathematical induction to prove that x=1nx2=n(n+1)(2n+1)6x=1nx2=n(n+1)(2n+1)6 size 12{ Sum cSub { size 8{x=1} } cSup { size 8{n} } {x rSup { size 8{2} } = { {n \( n+1 \) \( 2n+1 \) } over {6} } } } {}.

Exercise 3

In a room with n n people ( n 2 ) (n2), every person shakes hands once with every other person. Prove that there are n2n2n2n2 size 12{ { {n rSup { size 8{2} } - n} over {2} } } {} handshakes.

Exercise 4

Find and prove a formula for x=1nx3x=1nx3 size 12{ Sum cSub { size 8{x=1} } cSup { size 8{n} } {x rSup { size 8{3} } } } {}.

Hint:

You will have to play with it for a while to find the formula. Just write out the first four or five terms, and see if you notice a pattern. Of course, that won’t prove anything: that’s what the induction is for!

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks