Call Now to Set Up Tutoring:

(888) 888-0446

Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

Create An Account

All Algebra II Resources

10 Diagnostic Tests 630 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #82 : Summations And Sequences

Determine the sum of: 2+7+12+...+217+222

Possible Answers:

5290

4840

5040

5020

4960

Correct answer:

5040

Explanation:

Write the formula to determine the sum of an arithmetic series.

$S= n(\frac{a_1+a_n}{2})$

where is the number of terms, a_1 is the first term, and a_n is the last term.

a_1=2 , a_n =222

Use the following formula to determine how many terms are in this series.

a_1+(n-1)d =a_n

The term is the common difference. Since the numbers are spaced five units, d=5 .

Substitute the known values and solve for n.

2+(n-1)5 =222

Subtract two from both sides, and distribute the five through the binomial.

5n-5 =220

Add five on both sides.

5n-5 +5=220+5

5n=225

Divide by five.

n=45

Plug this value and the other givens to the sum formula to determine the sum.

$S= n(\frac{a_1+a_n}{2}) = 45(\frac{2+222}{2}) = 45(112) = 5040$

The answer is: 5040

Report an Error

Example Question #2751 : Algebra Ii

If the first term is 4, and the common difference is 3, what is the formula for the sequence?

Possible Answers:

a_n=10n-3

a_n=4n-3

a_n=3n+1

a_n=4n+2

a_n=4n-1

Correct answer:

a_n=3n+1

Explanation:

This represents an arithmetic sequence. Write the formula.

a_n=a_1+(n-1)d

Substitute the first term and the common difference in the formula.

a_n=4+(n-1)3

Simplify the terms.

a_n=4+3x-3

The answer is: a_n=3n+1

Report an Error

Example Question #22 : Arithmetic Series

Given the sequence: [5,11,17,...] , what is the hundredth term?

Possible Answers:

599

603

587

593

605

Correct answer:

599

Explanation:

Write the formula for the arithmetic sequence.

a_n = a_1+ d(n-1)

The first term is: a_1=5

The common difference is the same for each term, which is increasing by six every term: d=6

Substitute and simplify the formula.

a_n = 5+ 6(n-1)

a_n = 5+6n-6

a_n =6n-1

To find the hundredth term, plug in n=100 .

$a_{100} =6(100)-1 = 599$

The answer is: 599

Report an Error

Example Question #23 : Arithmetic Series

A sequence begins as follows:

8, 24, 72, 216,...

Which statement is true?

Possible Answers:

The sequence may be arithmetic.

The sequence cannot be arithmetic or geometric.

The sequence may be geometric.

Correct answer:

The sequence may be geometric.

Explanation:

An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the differences between each term and the previous term is not constant from term to term:

72 - 24 = 48

24 - 16= 8

The sequence cannot be arithmetic.

A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratio of each term to the previous one is the same:

$\frac{24}{8} = 3$

$\frac{72}{24} = 3$

$\frac{216}{72} = 3$

The sequence could be geometric.

Report an Error

Example Question #2752 : Algebra Ii

For the sequence of numbers $\left \{ 14, 19, 24, 29, ...\right \}$ , find the following:

a) An explicit formula for the nth term a_n

b) $a_{1017}-a_{101}$

Possible Answers:

a) a_n = 9 +5n

b) $a_{1017}-a_{101}=4580$

a) a_n = 5n

b) $a_{1017}-a_{101}=4589$

a) a_n =9+5(n-1)

b) $a_{1017}-a_{101} = 4589$

a) a_n = 14+5n

b) $a_{1017}-a_{101}=4580$

a) a_n = 14 +5(n+1)

b) $a_{1017}-a_{101}=501$

Correct answer:

a) a_n = 9 +5n

b) $a_{1017}-a_{101}=4580$

Explanation:

a) The general formula for an arithmetic sequence of terms $\left \{ a_1, a_2, ...a_n \right \}$ is written,

a_n = a_1 +d(n-1)

a_1 is the first term in the sequence, and is the common difference, which is just the difference between any two adjacent terms in the sequence.

$d= a_n -a_{n-1}$

For the sequence to be a true arithmetic sequence, the common difference must be the same for any two adjacent terms in the sequence.

For our sequence:

$\left \{ 14, 19, 24, 29, ...\right \}$

d=19-14 =5

d = 24-19 = 5

d = 29-14 = 5

and so on...

An explicit formula can now be written,

a_n = 14 +5(n-1)

a_n = 9+5n

We can now test this formula for the first few terms (columns 1-3 in the table below).

b) In the table below, columns 4-5 show the calculations for the $a_{1017}$ term and the $a_{101}$ term. The difference is simply 4580 .

New chart

Report an Error

Example Question #31 : Arithmetic Series

An arithmetic sequence begins as follows:

$a_{1} = 13$

$a_{2}= 19$

Which of the following gives the definition of its th term?

Possible Answers:

$a_{n} =13 \cdot 6 ^{n }$

$a_{n}= 6n+7$

$a_{n} =13 \cdot 6 ^{n-1}$

$a_{n}= 6n+13$

$a_{n}= 6n+1$

Correct answer:

$a_{n}= 6n+7$

Explanation:

The th term of an arithmetic sequence is

$a_{n}= a_{1}+ (n-1)d$ ,

where $a_{1}$ is its initial term and is the common difference between the terms.

$a_{1} = 13$ ,

and

$d = a_{2} - a_{1} = 19- 13 = 6$

Substituting, the expression becomes

$a_{n}= 13 + (n-1)6$

Simplify this:

$a_{n}= 13 + 6n-6$

$a_{n}= 6n+7$

Report an Error

Example Question #31 : Arithmetic Series

List the first five terms $\left \{a_1, a_2, a_3, a_4, a_5 \right \}$ of an infinite sequence of consecutive odd integers such that the sum of the 2nd plus 3-times the 1st is 1-less than the sum of the 3rd integer plus 2-times the 5th integer. Write an explicit formula for the arithmetic sequence.

Possible Answers:

a) $\left \{ 9,11,13,15,17\right \}$

b) a_n = 7+2n

a) $\left \{ 17,19,21,23,25\right \}$

b) a_n = 15+2n

a) $\left \{ 17,19,21,23,25\right \}$

b) a_n = 17+2(n+1)

a) $\left \{ 23,25,27,29,31\right \}$

b) a_n = 23+2(n+1)

a) $\left \{ 23,25,27,29,31\right \}$

b) a_n = 21+2n

Correct answer:

a) $\left \{ 17,19,21,23,25\right \}$

b) a_n = 15+2n

Explanation:

a) First we will start with creating the list of consecutive odd integers with the required relationships satisfied.

Start by calling the first of these five integers . The next odd integer will therefore be n +2 , and the third will be n+4 and so on... We can list them in a table to keep track:

Consecutive iintegers table

Now we need to translate the conditions given in the text of the problem into a mathematical expression. Write it out piece-by-piece. First let's right out these two statements separately,

...the sum of the 2nd plus 3-times the first...

(n+2)+3n

...the sum of the 3rd integer plus 2-times the 5th integer...

2(n+8)+(n+4)

In the text of the problem the first quantity "is one less than" the other quantity. In other words, the quantity (n+2)+3n is one less than 2(n+8)+(n+4) . Putting everything together we obtain:

(n+2)+3n=2(n+8)+(n+4)-1

Now solve for ,

4n+2 =3n+19

n = 17

Now we can use this solution to find the other four integers:

Consecutive integer problem solutions for n

b) Now that we have the first 5 terms, $\left \{ 17, 19, 21, 23, 25 \right \}$ , we can find an explicit formula for the arithmetic sequence. The general form of this is equation is,

a_n = a_1 +d(n-1)

Where is the common difference between adjacent terms and a_1 is the first term in the sequence. Because each adjacent term has a difference of , we have d=2 .

a_n = 17+2(n-1)

a_n = 15+2n

Report an Error

Example Question #32 : Arithmetic Series

A sequence begins as follows:

10, -11, -32, -53, ...

Which statement is true?

Possible Answers:

The sequence cannot be arithmetic or geometric.

None of these

The sequence may be geometric.

The sequence may be arithmetic and geometric.

The sequence may be arithmetic.

Correct answer:

The sequence may be arithmetic.

Explanation:

$\frac{-11}{10} = - \frac{11}{10}$

$\frac{-32}{-11} = \frac{32}{11}$

These ratios can be immediately seen to be unequal as they are of different sign.

An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the difference between each term and the previous term is the same:

-11 - 10 = -21

-32 - (-11) = -21

-53 - (-32) = -21

The sequence could be arithmetic.

Report an Error

Example Question #2751 : Algebra Ii

Which of the following infinite series has a finite sum?

Possible Answers:

$\sum_{n=0}^{\infty } \left ( \frac{1}{2} \right )^{n}$

$\sum_{n=0}^{\infty } 3*5^{n}$

$\sum_{n=0}^{\infty } \left ( \frac{7}{3} \right )^{n}$

$\sum_{n=0}^{\infty } 1.15^{n}$

$\sum_{n=0}^{\infty } \left ( \frac{3}{2} \right )^{n}$

Correct answer:

$\sum_{n=0}^{\infty } \left ( \frac{1}{2} \right )^{n}$

Explanation:

For an infinite series to have a finite sum, the exponential term (the term being raised to the power of in each term of the series) must be between and . Otherwise, each term is larger than the previous term, causing the overall sum to grow without bounds towards infinity.

Report an Error

Example Question #2 : Infinite Series

Evaluate:

$\sum_{i = 0}^{\infty} \left (- \frac{3}{5} \right )^{i}$

Possible Answers:

The series diverges.

$-\frac{3}{8}$

$\frac{5}{8}$

$\frac{3}{8}$

$-\frac{5}{8}$

Correct answer:

$\frac{5}{8}$

Explanation:

The sum of an infinite series $\sum_{i = 0}^{\infty}a^{i}$ , where |a| < 1 , can be calculated as follows:

$\sum_{i = 0}^{\infty} a^{i} = \frac{1 }{1- a}$

Setting $a = - \frac{3}{5}$ :

$\sum_{i = 0}^{\infty} \left (- \frac{3}{5} \right )^{i} = \frac{1 }{1- \left (- \frac{3}{5} \right )}$

$= \frac{1 }{1+ \frac{3}{5} }$

$= \frac{1 }{ \frac{8}{5} }$

$= \frac{5}{8}$

Report an Error

View Algebra 2 Tutors

Therar
Certified Tutor

Beirut Arab University, Doctor of Philosophy, Mathematics. Saint Joseph University of Beirut, Master of Science, Counselor Ed...

View Algebra 2 Tutors

Abe
Certified Tutor

CUNY Brooklyn College, Bachelor of Science, Accounting and Business Management. CUNY Graduate School and University Center, M...

View Algebra 2 Tutors

Leonard
Certified Tutor

Florida State University, Bachelor of Science, Environmental Science.

All Algebra II Resources

10 Diagnostic Tests 630 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Reading Tutors in Philadelphia, English Tutors in Phoenix, French Tutors in Phoenix, GMAT Tutors in Miami, Statistics Tutors in Los Angeles, SSAT Tutors in New York City, Calculus Tutors in Dallas Fort Worth, SSAT Tutors in Dallas Fort Worth, Physics Tutors in New York City, Algebra Tutors in Chicago

Popular Courses & Classes

GMAT Courses & Classes in Washington DC, GMAT Courses & Classes in Los Angeles, Spanish Courses & Classes in Dallas Fort Worth, SAT Courses & Classes in Boston, ISEE Courses & Classes in San Diego, ISEE Courses & Classes in Philadelphia, ISEE Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Boston, Spanish Courses & Classes in Atlanta, MCAT Courses & Classes in Washington DC

Popular Test Prep

SAT Test Prep in Los Angeles, GMAT Test Prep in Seattle, GMAT Test Prep in Atlanta, GMAT Test Prep in San Diego, LSAT Test Prep in San Francisco-Bay Area, ACT Test Prep in Philadelphia, SAT Test Prep in Philadelphia, MCAT Test Prep in Washington DC, GRE Test Prep in Seattle, MCAT Test Prep in Miami

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #82 : Summations And Sequences

Example Question #2751 : Algebra Ii

Example Question #22 : Arithmetic Series

Example Question #23 : Arithmetic Series

Example Question #2752 : Algebra Ii

Example Question #31 : Arithmetic Series

Example Question #31 : Arithmetic Series

Example Question #32 : Arithmetic Series

Example Question #2751 : Algebra Ii

Example Question #2 : Infinite Series

All Algebra II Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: