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For the following exercises, write the first four terms of the sequence.

a n = n ! n 2

First four terms: 1 , 1 2 , 2 3 , 3 2

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a n = 3 n ! 4 n !

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a n = n ! n 2 n 1

First four terms: 1 , 2 , 6 5 , 24 11

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a n = 100 n n ( n 1 ) !

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Graphical

For the following exercises, graph the first five terms of the indicated sequence

a n = { 4 + n 2 n if  n  is even 3 + n if  n  is odd

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a 1 = 2 ,   a n = ( a n 1 + 1 ) 2

Graph of a scattered plot with points at (1, 2), (2, 1), (3, 0), (4, 1), and (5, 0). The x-axis is labeled n and the y-axis is labeled a_n.
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a n = 1 ,   a n = a n 1 + 8

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a n = ( n + 1 ) ! ( n 1 ) !

Graph of a scattered plot with labeled points: (1, 2), (2, 6), (3, 12), (4, 20), and (5, 30). The x-axis is labeled n and the y-axis is labeled a_n.
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For the following exercises, write an explicit formula for the sequence using the first five points shown on the graph.

For the following exercises, write a recursive formula for the sequence using the first five points shown on the graph.

Graph of a scattered plot with labeled points: (1, 6), (2, 7), (3, 9), (4, 13), and (5, 21). The x-axis is labeled n and the y-axis is labeled a_n.

a 1 = 6 ,   a n = 2 a n 1 5

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Technology

Follow these steps to evaluate a sequence defined recursively using a graphing calculator:

  • On the home screen, key in the value for the initial term a 1 and press [ENTER] .
  • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes [2ND] ANS for the previous term a n 1 . Press [ENTER] .
  • Continue pressing [ENTER] to calculate the values for each successive term.

For the following exercises, use the steps above to find the indicated term or terms for the sequence.

Find the first five terms of the sequence a 1 = 87 111 ,   a n = 4 3 a n 1 + 12 37 . Use the> Frac feature to give fractional results.

First five terms: 29 37 , 152 111 , 716 333 , 3188 999 , 13724 2997

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Find the 15 th term of the sequence a 1 = 625 ,   a n = 0.8 a n 1 + 18.

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Find the first five terms of the sequence a 1 = 2 ,   a n = 2 [ ( a n 1 ) 1 ] + 1.

First five terms: 2 , 3 , 5 , 17 , 65537

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Find the first ten terms of the sequence a 1 = 8 ,   a n = ( a n 1 + 1 ) ! a n 1 ! .

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Find the tenth term of the sequence a 1 = 2 ,   a n = n a n 1

a 10 = 7 , 257 , 600

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Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following.

  • In the home screen, press [2ND] LIST .
  • Scroll over to OPS and choose “seq(” from the dropdown list. Press [ENTER] .
  • In the line headed “Expr:” type in the explicit formula, using the [ X,T , θ , n ] button for n
  • In the line headed “Variable:” type in the variable used on the previous step.
  • In the line headed “start:” key in the value of n that begins the sequence.
  • In the line headed “end:” key in the value of n that ends the sequence.
  • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms.

Using a TI-83, do the following.

  • In the home screen, press [2ND] LIST .
  • Scroll over to OPS and choose “seq(” from the dropdown list. Press [ENTER] .
  • Enter the items in the order “Expr” , “Variable” , “start” , “end” separated by commas. See the instructions above for the description of each item.
  • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms.

For the following exercises, use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary.

List the first five terms of the sequence a n = 28 9 n + 5 3 .

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List the first six terms of the sequence a n = n 3 3.5 n 2 +   4.1 n 1.5 2.4 n .

First six terms: 0.042 , 0.146 , 0.875 , 2.385 , 4.708

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List the first five terms of the sequence a n = 15 n ( 2 ) n 1 47

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List the first four terms of the sequence a n = 5.7 n + 0.275 ( n 1 ) !

First four terms: 5.975 , 32.765 , 185.743 , 1057.25 , 6023.521

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List the first six terms of the sequence a n = n ! n .

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Extensions

Consider the sequence defined by a n = 6 8 n . Is a n = 421 a term in the sequence? Verify the result.

If a n = 421 is a term in the sequence, then solving the equation 421 = 6 8 n for n will yield a non-negative integer. However, if 421 = 6 8 n , then n = 51.875 so a n = 421 is not a term in the sequence.

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What term in the sequence a n = n 2 + 4 n + 4 2 ( n + 2 ) has the value 41 ? Verify the result.

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Find a recursive formula for the sequence 1 ,   0 ,   1 ,   1 ,   0 ,   1 ,   1 ,   0 ,   1 ,   1 ,   0 ,   1 ,   1 ,   ...   . ( Hint : find a pattern for a n based on the first two terms.)

a 1 = 1 , a 2 = 0 , a n = a n 1 a n 2

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Calculate the first eight terms of the sequences a n = ( n + 2 ) ! ( n 1 ) ! and b n = n 3 + 3 n 2 + 2 n , and then make a conjecture about the relationship between these two sequences.

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Prove the conjecture made in the preceding exercise.

( n + 2 ) ! ( n 1 ) ! = ( n + 2 ) · ( n + 1 ) · ( n ) · ( n 1 ) · ... · 3 · 2 · 1 ( n 1 ) · ... · 3 · 2 · 1 = n ( n + 1 ) ( n + 2 ) = n 3 + 3 n 2 + 2 n

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Questions & Answers

find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
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
Â
Thomas
It is the  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
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
Jhon Reply
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
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
Practice Key Terms 8

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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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