<< Chapter < Page Chapter >> Page >

Write the first five terms of the sequence defined by the recursive formula.

a 1 = 2 a n = 2 a n 1 + 1 , for  n 2

{ 2 ,   5 ,   11 ,   23 ,   47 }

Got questions? Get instant answers now!

Given a recursive formula with two initial terms, write the first n terms of a sequence.

  1. Identify the initial term, a 1 , which is given as part of the formula.
  2. Identify the second term, a 2 , which is given as part of the formula.
  3. To find the third term, substitute the initial term and the second term into the formula. Evaluate.
  4. Repeat until you have evaluated the n th term.

Writing the terms of a sequence defined by a recursive formula

Write the first six terms of the sequence defined by the recursive formula.

a 1 = 1 a 2 = 2 a n = 3 a n 1 + 4 a n 2 , for  n 3

The first two terms are given. For each subsequent term, we replace a n 1 and a n 2 with the values of the two preceding terms.

n = 3 a 3 = 3 a 2 + 4 a 1 = 3 ( 2 ) + 4 ( 1 ) = 10 n = 4 a 4 = 3 a 3 + 4 a 2 = 3 ( 10 ) + 4 ( 2 ) = 38 n = 5 a 5 = 3 a 4 + 4 a 3 = 3 ( 38 ) + 4 ( 10 ) = 154 n = 6 a 6 = 3 a 5 + 4 a 4 = 3 ( 154 ) + 4 ( 38 ) = 614

The first six terms are {1,2,10,38,154,614} . See [link] .

Graph of a scattered plot with labeled points: (1, 1), (2, 2), (3, 10), (4, 38), (5, 154) and (6, 614). The x-axis is labeled n and the y-axis is labeled a_n.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Write the first 8 terms of the sequence defined by the recursive formula.

a 1 = 0 a 2 = 1 a 3 = 1 a n = a n 1 a n 2 + a n 3 , for  n 4

{ 0 ,   1 ,   1 ,   1 ,   2 ,   3 ,   5 2 ,   17 6 } .

Got questions? Get instant answers now!

Using factorial notation

The formulas for some sequences include products of consecutive positive integers. n factorial , written as n ! , is the product of the positive integers from 1 to n . For example,

4 ! = 4 3 2 1 = 24 5 ! = 5 4 3 2 1 = 120

An example of formula containing a factorial is a n = ( n + 1 ) ! . The sixth term of the sequence can be found by substituting 6 for n .

a 6 = ( 6 + 1 ) ! = 7 ! = 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5040

The factorial of any whole number n is n ( n 1 ) ! We can therefore also think of 5 ! as 5 4 ! .

Factorial

n factorial    is a mathematical operation that can be defined using a recursive formula. The factorial of n , denoted n ! , is defined for a positive integer n as:

0 ! = 1 1 ! = 1 n ! = n ( n 1 ) ( n 2 ) ( 2 ) ( 1 ) , for  n 2

The special case 0 ! is defined as 0 ! = 1.

Can factorials always be found using a calculator?

No. Factorials get large very quickly—faster than even exponential functions! When the output gets too large for the calculator, it will not be able to calculate the factorial.

Writing the terms of a sequence using factorials

Write the first five terms of the sequence defined by the explicit formula a n = 5 n ( n + 2 ) ! .

Substitute n = 1 , n = 2 , and so on in the formula.

n = 1 a 1 = 5 ( 1 ) ( 1 + 2 ) ! = 5 3 ! = 5 3 · 2 · 1 = 5 6 n = 2 a 2 = 5 ( 2 ) ( 2 + 2 ) ! = 10 4 ! = 10 4 · 3 · 2 · 1 = 5 12 n = 3 a 3 = 5 ( 3 ) ( 3 + 2 ) ! = 15 5 ! = 15 5 · 4 · 3 · 2 · 1 = 1 8 n = 4 a 4 = 5 ( 4 ) ( 4 + 2 ) ! = 20 6 ! = 20 6 · 5 · 4 · 3 · 2 · 1 = 1 36 n = 5 a 5 = 5 ( 5 ) ( 5 + 2 ) ! = 25 7 ! = 25 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5 1 , 008

The first five terms are { 5 6 , 5 12 , 1 8 , 1 36 , 5 1,008 } .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Write the first five terms of the sequence defined by the explicit formula a n = ( n + 1 ) ! 2 n .

The first five terms are { 1 ,   3 2 ,   4 ,   15 ,   72 } .

Got questions? Get instant answers now!

Access this online resource for additional instruction and practice with sequences.

Key equations

Formula for a factorial 0 ! = 1 1 ! = 1 n ! = n ( n 1 ) ( n 2 ) ( 2 ) ( 1 ) , for  n 2

Key concepts

  • A sequence is a list of numbers, called terms, written in a specific order.
  • Explicit formulas define each term of a sequence using the position of the term. See [link] , [link] , and [link] .
  • An explicit formula for the n th term of a sequence can be written by analyzing the pattern of several terms. See [link] .
  • Recursive formulas define each term of a sequence using previous terms.
  • Recursive formulas must state the initial term, or terms, of a sequence.
  • A set of terms can be written by using a recursive formula. See [link] and [link] .
  • A factorial is a mathematical operation that can be defined recursively.
  • The factorial of n is the product of all integers from 1 to n See [link] .

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 8

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask