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Understanding the Summation Notation: Single and Double Sums, Double Index Notation, Schemes and Mind Maps of Pre-Calculus

An explanation of the summation notation, focusing on single and double sums and double index notation. It includes examples and rules for evaluating sums. Useful for students in mathematics, engineering, and physics.

What you will learn

  • What is the difference between single and double sums in summation notation?
  • What is the purpose of double index notation in representing data in tables or matrices?
  • How do you evaluate a double sum using double index notation?

Typology: Schemes and Mind Maps

2021/2022

Uploaded on 09/12/2022

conney
conney 🇺🇸

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THESUMMATIONSYMBOL
Summary
1.Simplesum..............................................................................................................................1
2.Doublesum..............................................................................................................................5
3.Doubleindex............................................................................................................................5
1. Simplesum
ThesymbolΣ(sigma)isgenerallyusedtodenoteasumofmultipleterms.Thissymbol
isgenerallyaccompaniedbyanindexthatvariestoencompassalltermsthatmustbe
consideredinthesum.
Forexample,thesumoffirstwholenumberscanberepresentedinthefollowing
manner:

 123.
Moregenerally,theexpression
 representsthesumofnterms
⋯..
Example1
Given3,5,6,27.
Evaluate
 and
 .
Solution:
Inthefirstsum,theindex"i"variesfrom1to5.Wemustthereforeincludethe5
termsinthesum.

 3562723
pf3
pf4
pf5

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THE SUMMATION SYMBOL

Summary

  1. Simple sum .............................................................................................................................. 1
  2. Double sum.............................................................................................................................. 5
  3. Double index............................................................................................................................ 5

1. Simple sum

The symbol Σ (sigma) is generally used to denote a sum of multiple terms. This symbol is generally accompanied by an index that varies to encompass all terms that must be considered in the sum.

For example, the sum of ݊ first whole numbers can be represented in the following manner:

݅෍

௡ ௜ୀଵ

More generally, the expression ∑ ௡௜ୀଵ ݔ௜represents the sum of n terms

ݔଵ ݔ ൅ଶ ݔ ൅ଷ ൅ ⋯. ൅ ݔ௡.

Example 1

Given ݔଵ ൌ 3, ݔଶ ൌ 5, ݔଷ ൌ 6, ݔସ ݔ ݀݊ܽ 2 ൌ (^) ହ ൌ 7.

Evaluate ∑^ ହ௜ୀଵ ݔ௜and ∑^ ସ௜ୀଶ ݔ௜.

Solution :

In the first sum, the index " i " varies from 1 to 5. We must therefore include the 5 terms in the sum.

ݔ ෍௜

ହ ௜ୀଵ

In the second case, the index " i "varies from 2 to 4. Only the terms ݔଶ ݔ ,ଷ and ݔସ must therefore be considered.

ݔ ෍௜

ସ ௜ୀଶ

When we use the summation symbol, it is useful to remember the following rules:

ݔ ܿ෍ (^) ௜

௡ ௜ୀଵ

௡ ௜ୀଵ ܿ෍

௡ ௜ୀଵ

௡ ௜ୀଵ

௡ ௜ୀଵ

௡ ௜ୀଵ Example 2

Given ݔଵ ൌ 3, ݔଶ ൌ 5, ݔଷ ൌ 6, ݔସ ݔ ݀݊ܽ 2 ൌ (^) ହ ൌ 7 et ݕଵ ൌ 2, ݕଶ ൌ 8, ݕଷ ൌ 3,

ݕସ ݕ ݀݊ܽ 1 ൌ (^) ହ ൌ 6.

Verify the three preceding rules with the following sums :

ܽሻ ෍ 4ݔ (^) ௜

ହ ௜ୀଵ

ܾ ሻ ෍ 4

ହ ௜ୀଵ

ܿ ሻ ෍ሺݔ (^) ௜ ݕ ൅௜ ሻ

ହ ௜ୀଵ

nor the expression

ݔ ෍௜ ݕ௜

௡ ௜ୀଵ with

ݔ ൭෍௜

௡ ௜ୀଵ

௡ ௜ୀଵ

Example 3

Given ݔଵ ൌ 3, ݔଶ ൌ 5, ݔଷ ൌ 6, ݔସ ݔ ݀݊ܽ 2 ൌ (^) ହ ൌ 7 and ݕଵ ൌ 2, ݕଶ ൌ 8, ݕଷ ൌ 3,

ݕସ ݕ ݀݊ܽ 1 ൌ (^) ହ ൌ 6.

a)

ݔ ෍௜ଶ^ ݔ ൌଵଶ^ ݔ ൅ଶଶ^ ݔ ൅ଷଶ^ ݔ ൅ସଶ^ ݔ ൅ହଶ^ ൌ

ହ ௜ୀଵ

3 ଶ^ ൅ 5 ଶ^ ൅ 6 ଶ^ ൅ 2 ଶ^ ൅ 7 ଶ^ ൌ 123

and

ݔ ൭෍௜

ହ ௜ୀଵ

ଶ ൌ ሺ3 ൅ 5 ൅ 6 ൅ 2 ൅ 7ሻଶ^ ൌ 23 ଶ^ ൌ 52് 123

b)

ݔ ෍௜ ݕ௜ ݔ ൌଵ ݕଵ ݔ ൅ଶ ݕଶ ݔ ൅ଷ ݕଷ ݔ ൅ସ ݕସ ݔ ൅ହ ݕହ

ହ ௜ୀଵ

and

ݔ ൭෍௜

௡ ௜ୀଵ

௡ ௜ୀଵ

2. Double sum

In certain situations, using a double sum may be necessary. You must then apply the definition successively.

Example

Given ݔ 1 ൌ 3, ݔ 2 ൌ 5, ݔ 3 ݕ ݀݊ܽ 1 ൌ 1 ൌ 2, ݕ 2 ൌ 4

We will use the index i for the terms of ݔ and index j for the terms of ݕ

෍ ෍ ݔ௜ ݕ௝

ଶ ௝ୀଵ

ଷ ௜ୀଵ

ଷ ୀଵ

3. Double index

To represent the data of a table or a matrix, we often use a double index notation, like ݔ௜௝ where the first index ( i ) corresponds to the number of the row where the data is located and the second ( j ) to the column. For example, the term ݔ 24 represents the data that is situated at the intersection of the 2 nd^ row and the 4 th^ column of the table or the matrix.

Example

Given

ݔଵଵ ൌ 4 ݔଵଶ ൌ 4 ݔଵଷ ൌ 1 ݔଵସ ൌ 5 ݔଶଵ ൌ 0 ݔଶଶ ൌ 3 ݔଶଷ ൌ 1 ݔଶସ ൌ 2 ݔଷଵ ൌ 1 ݔଷଶ ൌ 4 ݔଷଷ ൌ 2 ݔଷସ ൌ 3

To carry out the sum of the terms of a row, we must fix the index of that row and vary, for all possible values, the index of the column. For example:

∑ ସ௝ୀଵ ݔଵ௝ݔ ൌଵଵ ൅ ݔଵଶ ݔ ൅ଵଷ ݔ ൅ଵସ ൌ 2 ൅ 4 ൅ 1 ൅ 5 ൌ 12 (sum of the first row)

∑ ସ௝ୀଵ ݔଶ௝ݔ ൌଶଵ ൅ ݔଶଶ ݔ ൅ଶଷ ݔ ൅ଶସ ൌ 0 ൅ 3 ൅ 1 ൅ 2 ൌ 6 (sum of the 2 nd^ row)

To carry out the sum of the terms of a column, you must fix the index of this column and vary, for all possible values, the index of the row.