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i = 1 n 1 = n

Another simple arithmetic sequence is when a 1 = 1 and d = 1 , which is the sequence of positive integers:

a i = a 1 + d ( i - 1 ) = 1 + 1 · ( i - 1 ) = i { a i } = { 1 ; 2 ; 3 ; 4 ; 5 ; ... }

If we wish to sum this sequence from i = 1 to any positive integer n , we would write

i = 1 n i = 1 + 2 + 3 + ... + n

This is an equation with a very important solution as it gives the answer to the sum of positive integers.

Interesting fact

Mathematician, Karl Friedrich Gauss, discovered this proof when he was only 8 years old. His teacher had decided to give his class a problem which would distract them for the entire day by asking them to add all the numbers from 1 to 100. Young Karl realised how to do this almost instantaneously and shocked the teacher with the correct answer, 5050.

We first write S n as a sum of terms in ascending order:

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

We then write the same sum but with the terms in descending order:

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

We then add corresponding pairs of terms from equations [link] and [link] , and we find that the sum for each pair is the same, ( n + 1 ) :

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

We then have n -number of ( n + 1 ) -terms, and by simplifying we arrive at the final result:

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

Note that this is an example of a quadratic sequence.

General formula for a finite arithmetic series

If we wish to sum any arithmetic sequence, there is no need to work it out term-for-term. We will now determine the general formula to evaluate a finite arithmetic series. We start with the general formula for an arithmetic sequence and sum it from i = 1 to any positive integer n :

i = 1 n a i = i = 1 n [ a 1 + d ( i - 1 ) ] = i = 1 n ( a 1 + d i - d ) = i = 1 n [ ( a 1 - d ) + d i ] = i = 1 n ( a 1 - d ) + i = 1 n ( d i ) = i = 1 n ( a 1 - d ) + d i = 1 n i = ( a 1 - d ) n + d n 2 ( n + 1 ) = n 2 ( 2 a 1 - 2 d + d n + d ) = n 2 ( 2 a 1 + d n - d ) = n 2 [ 2 a 1 + d ( n - 1 ) ]

So, the general formula for determining an arithmetic series is given by

S n = i = 1 n [ a 1 + d ( i - 1 ) ] = n 2 [ 2 a 1 + d ( n - 1 ) ]

For example, if we wish to know the series S 20 for the arithmetic sequence a i = 3 + 7 ( i - 1 ) , we could either calculate each term individually and sum them:

S 20 = i = 1 20 [ 3 + 7 ( i - 1 ) ] = 3 + 10 + 17 + 24 + 31 + 38 + 45 + 52 + 59 + 66 + 73 + 80 + 87 + 94 + 101 + 108 + 115 + 122 + 129 + 136 = 1390

or, more sensibly, we could use equation [link] noting that a 1 = 3 , d = 7 and n = 20 so that

S 20 = i = 1 20 [ 3 + 7 ( i - 1 ) ] = 20 2 [ 2 · 3 + 7 ( 20 - 1 ) ] = 1390

This example demonstrates how useful equation [link] is.

Exercises

  1. The sum to n terms of an arithmetic series is S n = n 2 ( 7 n + 15 ) .
    1. How many terms of the series must be added to give a sum of 425?
    2. Determine the 6 th term of the series.
  2. The sum of an arithmetic series is 100 times its first term, while the last term is 9 times the first term. Calculate the number of terms in the series if the first term is not equal to zero.
  3. The common difference of an arithmetic series is 3. Calculate the values of n for which the n t h term of the series is 93, and the sum of the first n terms is 975.
  4. The sum of n terms of an arithmetic series is 5 n 2 - 11 n for all values of n . Determine the common difference.
  5. The sum of an arithmetic series is 100 times the value of its first term, while the last term is 9 times the first term. Calculate the number of terms in the series if the first term is not equal to zero.
  6. The third term of an arithmetic sequence is -7 and the 7 th term is 9. Determine the sum of the first 51 terms of the sequence.
  7. Calculate the sum of the arithmetic series 4 + 7 + 10 + + 901 .
  8. The common difference of an arithmetic series is 3. Calculate the values of n for which the n th term of the series is 93 and the sum of the first n terms is 975.

Questions & Answers

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
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