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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

factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
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where can I get indices
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I need matrices
Nasasira
hi
Raihany
Hi
Solomon
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Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
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you guys know any app with matrices?
Khay
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Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
yah
immy
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
how would this look as an equation?
Hayden
5x+x=45
Khay
Practice Key Terms 8

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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