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Mathematics Learning Centre

Sigma notation

Jackie Nicholas

c?2005 University of Sydney Mathematics Learning Centre, University of Sydney1

1 Sigma Notation

1.1 Understanding Sigma Notation

The symbol Σ (capital sigma) is often used as shorthand notation to indicate the sum of a number of similar terms. Sigma notation is used extensively in statistics. For example, suppose we weigh five children. We will denote their weights byx 1 ,x 2 ,x 3 x 4 andx 5

The sum of their weightsx

1 +x 2 +x 3 +x 4 +x 5 is written more compactly as 5 j=1 x j The symbol Σ means 'add up". Underneath Σ we seej= 1 and on top of it 5. This means thatjis replaced by whole numbers starting at the bottom number, 1, until the top number,5, is reached. Thus 5 j=2 x j =x 2 +x 3 +x 4 +x 5 and 4 j=2 x j =x 2 +x 3 +x 4

So the notation

n j=1 x j tells us: a.to add the scoresx j b.where to start:x 1 c.where to stop:x n (wherenis some number).

Now take the weights of the children to bex

1 = 10kg,x 2 = 12kg,x 3 = 14kg,x 4 = 8kg andx 5 = 11kg. Then the total weight (in kilograms) is 5 i=1 x i =x 1 +x 2 +x 3 +x 4 +x 5 = 10+12+14+8+11 =55. Notice that we have usediinstead ofjin the formula above. Thejis what we call a dummy variable - any letter can be used, ie, n j=1 x j n i=1 x i

Now let us find

4 i=1 2x i wherex 1 =2,x 2 =3,x 3 =-2 andx 4 =1. Mathematics Learning Centre, University of Sydney2 Again, starting withi= 1 we replace the expression 2x i with its value and add up the terms untili= 4 is reached. So, 4 i=1 2x i =2x 1 +2x 2 +2x 3 +2x 4 = 2(2) + 2(3) + 2(-2) + 2(1) =4+6-4+2 =8.

Similarly, let us find

3 k=1 (x k -4) wherex 1 =7,x 2 =4,x 3 =1. Here, 3 k=1 (x k -4) = (x 1 -4) + (x 2 -4)+(x 3 -4) =(7-4) + (4-4) + (1-4) = 3+0+(-3) =0.

Notice that this is different from

3 k=1 x k -4 wherex 1 =7,x 2 =4,x 3 =1.

In this case, we have,

3 k=1 x k -4=x 1 +x 2 +x 3 -4 = 7+4+1-4 =8. We use brackets to indicate what should be included in the sum. In the previous example, there were no brackets, so the '4" was not included in the sum.

Example:Write out in full:

5 k=1 x k

Solution:x

1 +x 2 +x 3 +x 4 +x 5

We also use sigma notation in the following way:

4 j=1 j 2 =1 2 +2 2 +3 2 +4 2 =30. This is the same principle: replacejin the expression (this timej 2 ) by whole numbers starting with 1 and ending with 4 , and add. Mathematics Learning Centre, University of Sydney3

1.1.1 Exercises

1.Evaluate

4 i=1 x i wherex 1 =5,x 2 =2,x 3 =3,x 4 =8.

2.Evaluate

n k=1 5x k wherex 1 = 10,x 2 = 14,x 3 =-2, andn=3.

3.Findμ=1

5 5 j=1 x j where thex 1 = 10kg,x 2 = 12kg,x 3 = 14kg,x 4 = 8kg andx 5 = 11kg are the weights of 5 children. (μis the mean weight of the children.)

4.Find the value of

3 i=1 (x i 2 wherex 1 = 105,x 2 = 100,x 3 = 95, andμ= 100.

1.2 Rules of summation

We will prove three rules of summation. These rules will allow us to evaluate formulae containing sigma notation more easily and allow us to derive equivalent formulae.

Rule 1:Ifcis a constant, then

n i=1 cx i =c n i=1 x i To see why Rule 1 is true, let"s start with the left hand side of this equation, n i=1 cx i =cx 1 +cx 2 +cx 3 +···+cx n =c(x 1 +x 2 +x 3 +···+x n =c n i=1 x i as required.

Suppose that

5 i=1 x i = 55 as in a previous example. Then 5 i=1 3x i =3 5 i=1 x i =3×55 = 165.

Rule 2:Ifcis a constant, then

n i=1 c=nc.

This rule looks a bit strange as there is no 'x

i ". The left hand side of this formula means 'sumc,ntimes". That is, n i=1 c= n c+c+···+c =n×c =nc.quotesdbs_dbs29.pdfusesText_35