Skip to content Skip to navigation Skip to collection information

Connexions

You are here: Home » Content » Advanced Algebra II: Activities and Homework » Sample Test: Sequences and 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 "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.
 

Sample Test: Sequences and Series

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

Summary: An updated version of the Sample Test: Sequences and Series module.

Exercise 1

The teacher sees the wishing star high in the night sky, and makes the mistake of wishing for whiteboard markers. The next day (let’s call it “day 1”), the marker fairy arrives and gives the teacher a marker that works—hooray! The day after that (“day 2”), the marker fairy gives the teacher three markers. The day after that, nine new markers…and so on…each day, three times as many new markers as the day before.

  • a. If n n is the day, and m m is the number of new markers the fairy brings that day, is the list of all m m numbers an arithmetic sequence, geometric sequence, or neither?
  • b. Give a “recursive definition” for the sequence: that is, a formula for m n + 1 m n + 1 based on m n m n .
  • c. Give an “explicit definition” for the sequence: that is, a formula that I can use to quickly find m n m n for any given n n, without finding all the previous m m terms.
  • d. On “day 30” (the end of the month) how many markers does the fairy bring?
  • e. After that “day 30” shipment, how many total markers has the fairy brought?

Exercise 2

You start a dot-com business. Like all dot-com businesses, it starts great, and then starts going downhill fast. Specifically, you make $10,000 the first day. Every day thereafter, you make $200 less than the previous day—so the second day you make $9,800, and the third day you make $9,600, and so on. You might think this pattern stops when you hit zero, but the pattern just keeps right on going—the day after you make $0, you lose $200, and the day after that you lose $400, and so on.

  • a. If n n is the day, and d d is the amount of money you gain on that day, is the list of all d d numbers an arithmetic sequence, geometric, or neither?
  • b. How much money do you make on the 33rd day?
  • c. On the day when you lose $1,000 in one day, you finally close up shop. What day is that?
  • d. Your accountant needs to figure out the total amount of money you made during the life of the business. Express this question in summation notation.
  • e. Now answer the question.

Exercise 3

Consider the series 23+29+227+281+...23+29+227+281+... size 12{ { {2} over {3} } + { {2} over {9} } + { {2} over {"27"} } + { {2} over {"81"} } + "." "." "." } {}

  • a. Is this an arithmetic series, a geometric series, or neither?
  • b. Write this series in summation notation (with a Σ Σ).
  • c. What is t 1 t 1 , and what is r r (if it is geometric) or d d (if it is arithmetic)?
  • d. What is the sum of the first 4 terms of this series?
  • e. What is the sum of the first n n terms of this series?

Exercise 4

Use induction to prove the following formula for all n n:

11×2+12×3+13×4+...+1n(n+1)= nn+111×2 size 12{ { {1} over {1 times 2} } } {}+12×3 size 12{ { {1} over {2 times 3} } } {}+13×4 size 12{ { {1} over {3 times 4} } } {}+...+1n(n+1) size 12{ { {1} over {n \( n+1 \) } } } {}=nn+1 size 12{ { {n} over {n+1} } } {}

Extra credit:

An arithmetic series starts with t 1 t 1 and goes up by d d each term for n n terms. Use the “arithmetic series trick” to find the general formula for the sum of this series, as a function of t 1 t 1 , n n, and d d.

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