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THE SUMMATION SYMBOL
Summary
- Simple sum .............................................................................................................................. 1
- Double sum.............................................................................................................................. 5
- 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 ݔrepresents 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 ∑^ ହୀଵ ݔand ∑^ ସୀଶ 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:
∑ ସୀଵ ݔଵݔ ൌଵଵ ݔଵଶ ݔ ଵଷ ݔ ଵସ ൌ 2 4 1 5 ൌ 12 (sum of the first row)
∑ ସୀଵ ݔଶݔ ൌ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.
