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THESUMMATIONSYMBOL

Summary1. Simplesum.............................................................................................................................. 1

2. Doublesum.............................................................................................................................. 5

3. Doubleindex............................................................................................................................ 5

1. Simplesum

The

symbol

(sigma)

is

generally

used

to

denote

a

sum

of

multiple

terms.

This

symbol

isgenerallyaccompaniedbyan indexthatvariestoencompassalltermsthatmustbe

consideredinthesum.

For example, the sum of firstwhole numbers canbe represented in the followingmanner:

1 2 3 . More

generally,

the

expression representsthesumofnterms . .

Example

1

Given 3, 5, 6, 2 7.Evaluate and .Solution:

In the firstsum,the index " i"varies from 1 to5.Wemust therefore includethe5

termsinthesum.

3 5 6 2 7 23

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Inthesecondcase,theindex"i"variesfrom2to4.Onlytheterms, andmustthereforebeconsidered.

5 6 2 13

Whenweusethesummationsymbol,itisusefultorememberthefollowingrules:

Example

2

Given 3, 5, 6, 2 7et 2, 8, 3,

1 6.

Verifythethreeprecedingruleswiththefollowingsums:

4

4

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Solution

:

4

4 4 4 4 4 4 3 4 5 4 6 4 2 4 7

92

and

4

4 3 5 6 2 7 4 23 92

b)

4

4 4 4 4 4 5 4 20

c)

3 2 5 8 6 3 2 1 7 6 43

And

3 5 72 8 6 23 20 43

Attention:

Wemustneitherconfoundtheexpression

with

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northeexpression

with

Example3

Given 3, 5, 6, 2 7and 2, 8, 3,

1 6.a)

3 5 6 2 7 123

and

3 5 6 2 7 23 52 123

b)

3 2 5 8 7 6

6 40 42 108and

3 5 7 2 8 6 23 20 460 108

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2.Doublesum

In certain situations,usingadouble summaybenecessary.Youmust thenapply the

definitionsuccessively.

Example

Given1 3,2 5,3 1 1 2,2 4Wewillusetheindexiforthetermsofandindexjforthetermsof

3 2 3 4 5 2 5 4 1 2 1 4 543.Doubleindex

Torepresentthedataofatableoramatrix,weoftenuseadoubleindexnotation,like where the first index (i)corresponds to thenumberof the rowwhere thedata islocatedandthesecond(j)tothecolumn.Forexample,theterm24representsthedatathatissituatedattheintersectionofthe2

ndrowandthe4

thcolumnofthetableorthe

matrix.

Example

Given

4 4 1 5 0 3 1 2 1 4 2 3

Tocarryoutthesumofthetermsofarow,wemustfixtheindexofthatrowandvary,

forallpossiblevalues,theindexofthecolumn.Forexample:

2 4 1 5 12(sumofthefirstrow) 0 3 1 2 6(sumofthe2ndrow)Tocarryoutthesumofthetermsofacolumn,youmustfixtheindexofthiscolumnand

vary,forallpossiblevalues,theindexoftherow.

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Forexample:

5 2 3 10(sumofthe4thcolumn)Tocarryout the sumofall termsof the table,youmustvaryboth indicesandusea

double

sum:

2 4 1 6 0 3 3 28