11.2 Arithmetic sequences  (Page 6/8)

 Page 6 / 8

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.

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
how do you get the (1.4427)^t in the carp problem?
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
hello
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
rotation by 80 of (x^2/9)-(y^2/16)=1
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
what is the standard form if the focus is at (0,2) ?
a²=4