Skip to content Skip to navigation Skip to collection information

Connexions

You are here: Home » Content » Advanced Algebra II: Teacher's Guide » Arithmetic and Geometric Series

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 "teacher's guide" 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 […]"

    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.
 

Arithmetic and Geometric Series

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

Summary: A teacher's guide to lecturing on arithmetic and geometric series.

Going over the homework, make sure to mention #3(e), an alternating series. You get that kind of alternation by throwing in a ( -1 ) n (-1 ) n or, in this case, ( -1 ) n - 1 (-1 ) n - 1 .

Last night’s homework ended with the series “all the even numbers between 50 and 100.” Some students may have written ∑n=126(48+2n)∑n=126(48+2n) size 12{ Sum cSub { size 8{n=1} } cSup { size 8{"26"} } { \( "48"+2n \) } } {}. Others may have written the answer differently. But one thing they probably all agree on is that adding it up would be a pain. If only there were...a shortcut!

Let’s consider the series 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 3+5+7+9+11+13+15+17. (Write that on the board.)

Exercise 1

What do we get if we add the first term to the last? Answer: 20. Modify your drawing on the board to look like this:

Figure 1
Examining the series by adding the first and last numbers together.

OK, what about the second term to the second-to-last? Hmm....20 again. Add to the drawing, and then keep adding until it looks like this:

Figure 2
Examining the series by adding the first and last numbers together some more.

Exercise 2

So, looking at that drawing, what does 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 3+5+7+9+11+13+15+17 add up to? Hopefully everyone can see that it adds up to four 20s, or 80.

Exercise 3

And this is the big one—will that trick work for all series? If so, why? If not, which series will it work for? Answer: It will work for all arithmetic series. The reason that the second pair added up the same as the first pair was that we went up by two on the left, and down by two on the right. As long as you go up by the same as you go down, the sum will stay the same—and this is just what happens for arithmetic series.

OK, what about geometric series? Write the following on the board:

2 + 6 + 18 + 54 + 162 + 486 + 1458 2+6+18+54+162+486+1458

Clearly the “arithmetic series trick” will not work here: 2 + 1458 2+1458 is not 6 + 486 6+486. We need a whole new trick. Here it comes. First, to the left of your equation, write S = S= so the board looks like:

S = 2 + 6 + 18 + 54 + 162 + 486 + 1458 S=2+6+18+54+162+486+1458

where S S is the mystery sum we’re looking for. Now, above that, write:

3 S = 3S=

ask the class what comes next. Can we just multiply each term by 3? (Yes, distributive property.) When you write this line, line up the numbers like this:

3 S = 6 + 18 + 54 + 162 + 486 + 1458 + 437 3S=6+18+54+162+486+1458+437

S = 2 + 6 + 18 + 54 + 162 + 486 + 1458 S=2+6+18+54+162+486+1458

But don’t go too fast on that step—make sure they see why, if S S is what we said, then 3 S 3S must be that!

Now, underline the second equation (as I did above), and then subtract the two equations. What do we get on the left of the equal sign? What do we get on the right? See how things cancel? See if you can get the class to tell you that...

2 S = 4374 − 2 2S=4374−2

So then S S is just 2186. They may want to verify this one on their calculators. Once again, however, the key is to understand why this trick always works for any Geometric series.

Homework:

“Homework: Arithmetic and Geometric Series”

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 | 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 ...

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