# 11.2 Arithmetic sequences  (Page 6/8)

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For the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence.

${a}_{n}=24-4n$

First five terms: $20,16,12,8,4.$

${a}_{n}=\frac{1}{2}n-\frac{1}{2}$

For the following exercises, write an explicit formula for each arithmetic sequence.

${a}_{n}=\left\{3,5,7,...\right\}$

${a}_{n}=1+2n$

${a}_{n}=\left\{32,24,16,...\right\}$

${a}_{n}=-105+100n$

${a}_{n}=1.8n$

${a}_{n}=\left\{-18.1,-16.2,-14.3,...\right\}$

${a}_{n}=\left\{15.8,18.5,21.2,...\right\}$

${a}_{n}=13.1+2.7n$

${a}_{n}=\left\{0,\frac{1}{3},\frac{2}{3},...\right\}$

${a}_{n}=\frac{1}{3}n-\frac{1}{3}$

${a}_{n}=\left\{-5,-\frac{10}{3},-\frac{5}{3},\dots \right\}$

For the following exercises, find the number of terms in the given finite arithmetic sequence.

There are 10 terms in the sequence.

${a}_{n}=\left\{1.2,1.4,1.6,...,3.8\right\}$

${a}_{n}=\left\{\frac{1}{2},2,\frac{7}{2},...,8\right\}$

There are 6 terms in the sequence.

## Graphical

For the following exercises, determine whether the graph shown represents an arithmetic sequence.

The graph does not represent an arithmetic sequence.

For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence.

${a}_{1}=0,d=4$

${a}_{1}=9;{a}_{n}={a}_{n-1}-10$

${a}_{n}=-12+5n$

## Technology

For the following exercises, follow the steps to work with the arithmetic sequence ${a}_{n}=3n-2$ using a graphing calculator:

• Press [MODE]
• Select SEQ in the fourth line
• Select DOT in the fifth line
• Press [ENTER]
• Press [Y=]
• $n\text{Min}\text{\hspace{0.17em}}$ is the first counting number for the sequence. Set $\text{\hspace{0.17em}}n\text{Min}=1$
• $u\left(n\right)\text{\hspace{0.17em}}$ is the pattern for the sequence. Set $\text{\hspace{0.17em}}u\left(n\right)=3n-2$
• $u\left(n\text{Min)}\text{\hspace{0.17em}}$ is the first number in the sequence. Set $\text{\hspace{0.17em}}u\left(n\text{Min)}=1$
• Press [2ND] then [WINDOW] to go to TBLSET
• Set $\text{\hspace{0.17em}}\text{TblStart}=1$
• Set $\text{\hspace{0.17em}}\Delta \text{Tbl}=1$
• Set Indpnt: Auto and Depend: Auto
• Press [2ND] then [GRAPH] to go to the TABLE

What are the first seven terms shown in the column with the heading $u\left(n\right)\text{?}$

$1,4,7,10,13,16,19$

Use the scroll-down arrow to scroll to $n=50.$ What value is given for $u\left(n\right)\text{?}$

Press [WINDOW] . Set $\text{\hspace{0.17em}}n\text{Min}=1,n\text{Max}=5,x\text{Min}=0,x\text{Max}=6,y\text{Min}=-1,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{Max}=14.\text{\hspace{0.17em}}$ Then press [GRAPH] . Graph the sequence as it appears on the graphing calculator.

For the following exercises, follow the steps given above to work with the arithmetic sequence ${a}_{n}=\frac{1}{2}n+5$ using a graphing calculator.

What are the first seven terms shown in the column with the heading $\text{\hspace{0.17em}}u\left(n\right)\text{\hspace{0.17em}}$ in the TABLE feature?

Graph the sequence as it appears on the graphing calculator. Be sure to adjust the WINDOW settings as needed.

## Extensions

Give two examples of arithmetic sequences whose 4 th terms are $9.$

Give two examples of arithmetic sequences whose 10 th terms are $206.$

Answers will vary. Examples: ${a}_{n}=20.6n$ and ${a}_{n}=2+20.4\mathrm{n.}$

Find the 5 th term of the arithmetic sequence $\left\{9b,5b,b,\dots \right\}.$

Find the 11 th term of the arithmetic sequence $\left\{3a-2b,a+2b,-a+6b\dots \right\}.$

${a}_{11}=-17a+38b$

At which term does the sequence $\left\{5.4,14.5,23.6,...\right\}$ exceed 151?

At which term does the sequence $\left\{\frac{17}{3},\frac{31}{6},\frac{14}{3},...\right\}$ begin to have negative values?

The sequence begins to have negative values at the 13 th term, ${a}_{13}=-\frac{1}{3}$

For which terms does the finite arithmetic sequence $\left\{\frac{5}{2},\frac{19}{8},\frac{9}{4},...,\frac{1}{8}\right\}$ have integer values?

Write an arithmetic sequence using a recursive formula. Show the first 4 terms, and then find the 31 st term.

Answers will vary. Check to see that the sequence is arithmetic. Example: Recursive formula: ${a}_{1}=3,{a}_{n}={a}_{n-1}-3.$ First 4 terms: $\begin{array}{ll}3,0,-3,-6\hfill & {a}_{31}=-87\hfill \end{array}$

Write an arithmetic sequence using an explicit formula. Show the first 4 terms, and then find the 28 th term.

The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
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.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
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
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
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?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations