11.2 Arithmetic sequences (Page 2/8)

Precalculus

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

How To

Given the first term and the common difference of an arithmetic sequence, find the first several terms.

Add the common difference to the first term to find the second term.
Add the common difference to the second term to find the third term.
Continue until all of the desired terms are identified.
Write the terms separated by commas within brackets.

Writing terms of arithmetic sequences

Write the first five terms of the arithmetic sequence with $a_{1} = 17$ and $d = - 3$ .

Adding $- 3$ is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term.

The first five terms are ${17, 14, 11, 8, 5}$

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List the first five terms of the arithmetic sequence with $a_{1} = 1$ and $d = 5$ .

${1, 6, 11, 16, 21}$

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How To

Given any the first term and any other term in an arithmetic sequence, find a given term.

Substitute the values given for $a_{1}, a_{n}, n$ into the formula $a_{n} = a_{1} + (n - 1) d$ to solve for $d .$
Find a given term by substituting the appropriate values for $a_{1}, n,$ and $d$ into the formula $a_{n} = a_{1} + (n - 1) d .$

Writing terms of arithmetic sequences

Given $a_{1} = 8$ and $a_{4} = 14$ , find $a_{5}$ .

The sequence can be written in terms of the initial term 8 and the common difference $d$ .

{8, 8 + d, 8 + 2 d, 8 + 3 d}

We know the fourth term equals 14; we know the fourth term has the form $a_{1} + 3 d = 8 + 3 d$ .

We can find the common difference $d$ .

\begin{array}{l} a_{n} = a_{1} + (n - 1) d \\ a_{4} = a_{1} + 3 d \\ a_{4} = 8 + 3 d & Write the fourth term of the sequence in terms of a_{1} and d . \\ 14 = 8 + 3 d & Substitute 14 for a_{4} . \\ d = 2 & Solve for the common difference . \end{array}

Find the fifth term by adding the common difference to the fourth term.

a_{5} = a_{4} + 2 = 16

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Given $a_{3} = 7$ and $a_{5} = 17$ , find $a_{2}$ .

$a_{2} = 2$

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Using recursive formulas for arithmetic sequences

Some arithmetic sequences are defined in terms of the previous term using a recursive formula . The formula provides an algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common difference. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given.

\begin{array}{l} a_{n} = a_{n - 1} + d & n \geq 2 \end{array}

A General Note

Recursive formula for an arithmetic sequence

The recursive formula for an arithmetic sequence with common difference $d$ is:

\begin{array}{l} a_{n} = a_{n - 1} + d & n \geq 2 \end{array}

How To

Given an arithmetic sequence, write its recursive formula.

Subtract any term from the subsequent term to find the common difference.
State the initial term and substitute the common difference into the recursive formula for arithmetic sequences.

Writing a recursive formula for an arithmetic sequence

Write a recursive formula for the arithmetic sequence .

{- 18, - 7, 4, 15, 26, …}

The first term is given as $−18$ . The common difference can be found by subtracting the first term from the second term.

d = −7 - (−18) = 11

Substitute the initial term and the common difference into the recursive formula for arithmetic sequences.

\begin{array}{l} a_{1} = - 18 \\ a_{n} = a_{n - 1} + 11, for n \geq 2 \end{array}

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Q&A

Do we have to subtract the first term from the second term to find the common difference?

No. We can subtract any term in the sequence from the subsequent term. It is, however, most common to subtract the first term from the second term because it is often the easiest method of finding the common difference.

Questions & Answers

what is f(x)=

Karim Reply

I don't understand

Joe

Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."

Thomas

Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)

Thomas

GREAT ANSWER THOUGH!!!

Darius

Thanks.

Thomas

It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â

Thomas

Now it shows, go figure?

Thomas

what is this?

unknown Reply

i do not understand anything

unknown

lol...it gets better

Darius

I've been struggling so much through all of this. my final is in four weeks 😭

Tiffany

this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts

Darius

thank you I have heard of him. I should check him out.

Tiffany

is there any question in particular?

Joe

I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.

Tiffany

Sure, are you in high school or college?

Darius

Hi, apologies for the delayed response. I'm in college.

Tiffany

how to solve polynomial using a calculator

Ef Reply

So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?

KARMEL Reply

The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26

Rima Reply

The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?

Rima

I done know

Joe

What kind of answer is that😑?

Rima

I had just woken up when i got this message

Joe

Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that

Rima

i have a question.

Abdul

how do you find the real and complex roots of a polynomial?

Abdul

@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up

Nare

This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1

Abdul

@Nare please let me know if you can solve it.

Abdul

I have a question

juweeriya

hello guys I'm new here? will you happy with me

mustapha

The average annual population increase of a pack of wolves is 25.

Brittany Reply

how do you find the period of a sine graph

Imani Reply

Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period

if not then how would I find it from a graph

Imani

by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.

you could also do it with two consecutive minimum points or x-intercepts

I will try that thank u

Imani

Case of Equilateral Hyperbola

Jhon Reply

Zander

Shella

f(x)=4x+2, find f(3)

Benetta

f(3)=4(3)+2 f(3)=14

lamoussa

Vedant

pre calc teacher: "Plug in Plug in...smell's good" f(x)=14

Devante

8x=40

Chris

Explain why log a x is not defined for a < 0

Baptiste Reply

the sum of any two linear polynomial is what

Esther Reply

divide simplify each answer 3/2÷5/4

Momo Reply

divide simplify each answer 25/3÷5/12

Momo

how can are find the domain and range of a relations

austin Reply

the range is twice of the natural number which is the domain

Morolake

A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?

Diddy Reply

6000

Robert

more than 6000

Robert

For Plan A to reach $27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional $10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...

Gilbert

...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...

Gilbert

can I see the picture

Zairen Reply

How would you find if a radical function is one to one?

Peighton Reply

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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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