<< Chapter < Page Chapter >> Page >

Evaluate k = 2 5 ( 3 k 1 ) .

38

Got questions? Get instant answers now!

Using the formula for arithmetic series

Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence    is a sequence in which the difference between any two consecutive terms is the common difference    , d . The sum of the terms of an arithmetic sequence is called an arithmetic series . We can write the sum of the first n terms of an arithmetic series as:

S n = a 1 + ( a 1 + d ) + ( a 1 + 2 d ) + ... + ( a n d ) + a n .

We can also reverse the order of the terms and write the sum as

S n = a n + ( a n d ) + ( a n 2 d ) + ... + ( a 1 + d ) + a 1 .

If we add these two expressions for the sum of the first n terms of an arithmetic series, we can derive a formula for the sum of the first n terms of any arithmetic series.

S n = a 1 + ( a 1 + d ) + ( a 1 + 2 d ) + ... + ( a n d ) + a n + S n = a n + ( a n d ) + ( a n 2 d ) + ... + ( a 1 + d ) + a 1 2 S n = ( a 1 + a n ) + ( a 1 + a n ) + ... + ( a 1 + a n )

Because there are n terms in the series, we can simplify this sum to

2 S n = n ( a 1 + a n ) .

We divide by 2 to find the formula for the sum of the first n terms of an arithmetic series.

S n = n ( a 1 + a n ) 2

Formula for the sum of the first n Terms of an arithmetic series

An arithmetic series    is the sum of the terms of an arithmetic sequence. The formula for the sum of the first n terms of an arithmetic sequence is

S n = n ( a 1 + a n ) 2

Given terms of an arithmetic series, find the sum of the first n terms.

  1. Identify a 1 and a n .
  2. Determine n .
  3. Substitute values for a 1 a n , and n into the formula S n = n ( a 1 + a n ) 2 .
  4. Simplify to find S n .

Finding the first n Terms of an arithmetic series

Find the sum of each arithmetic series.

  1. 5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32
  2. 20 + 15 + 10 +…+ −50
  3. k = 1 12 3 k 8
  1. We are given a 1 = 5 and a n = 32.

    Count the number of terms in the sequence to find n = 10.

    Substitute values for a 1 , a n  , and n into the formula and simplify.

      S n = n ( a 1 + a n ) 2 S 10 = 10 ( 5 + 32 ) 2 = 185
  2. We are given a 1 = 20 and a n = 50.

    Use the formula for the general term of an arithmetic sequence to find n .

    a n = a 1 + ( n 1 ) d 50 = 20 + ( n 1 ) ( 5 ) 70 = ( n 1 ) ( 5 ) 14 = n 1 15 = n

    Substitute values for a 1 , a n , n into the formula and simplify.

    S n = n ( a 1 + a n ) 2 S 15 = 15 ( 20 50 ) 2 = 225
  3. To find a 1 , substitute k = 1 into the given explicit formula.

    a k = 3 k 8   a 1 = 3 ( 1 ) 8 = 5

    We are given that n = 12. To find a 12 , substitute k = 12 into the given explicit formula.

      a k = 3 k 8 a 12 = 3 ( 12 ) 8 = 28

    Substitute values for a 1 , a n , and n into the formula and simplify.

      S n = n ( a 1 + a n ) 2 S 12 = 12 ( 5 + 28 ) 2 = 138
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use the formula to find the sum of each arithmetic series.

1 .4 + 1 .6 + 1 .8 + 2 .0 + 2 .2 + 2 .4 + 2 .6 + 2 .8 + 3 .0 + 3 .2 + 3 .4

26 .4

Got questions? Get instant answers now!

13 + 21 + 29 +  + 69

328

Got questions? Get instant answers now!

k = 1 10 5 6 k

−280

Got questions? Get instant answers now!

Solving application problems with arithmetic series

On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile. After 8 weeks, what will be the total number of miles she has walked?

This problem can be modeled by an arithmetic series with a 1 = 1 2 and d = 1 4 . We are looking for the total number of miles walked after 8 weeks, so we know that n = 8 , and we are looking for S 8 . To find a 8 , we can use the explicit formula for an arithmetic sequence.

a n = a 1 + d ( n 1 ) a 8 = 1 2 + 1 4 ( 8 1 ) = 9 4

We can now use the formula for arithmetic series.

  S n = n ( a 1 + a n ) 2    S 8 = 8 ( 1 2 + 9 4 ) 2 = 11

She will have walked a total of 11 miles.

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

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

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask