Summation Notation Review
AGEC 317
The use of subscripted variables is common in economics. Subscripting variables
helps in (a) designating the location of variables, such as the location within a matrix, and
(b) allows for shorthand notation of terms such as in matrix manipulation.
1.
Summation notation makes use of the Greek letter sigma, ∑. To express the
sum of a series of variables, x1, x2, x3, and x4, we write the summation as
4
x1  x 2  x 3  x 4   x i .
i 1
Expand the following summation expressions.
3
(a )
 2x
k
k 1
3
( b)
 a (x
i
i 1
i
 1)
3
( c)
b x
i 1
2.
3.
i
i
Rewrite the following in ∑ notation.
(a )
x1  x 2  x 3  x 4  x 5
( b)
1
1
( c)
1
1
 3
h int x o  1
2
x
x
x
1
a 1 ( x 1  1)  a 2 ( x 22  1)
(d )
a1x 2  a 2 x 3  a 3 x 4

Show the following properties are true. To show let n = 3, this is not proving the
properties hold, but will show the properties.
(a )
n
n
i 1
i 1
 ax i  a  x i
n
( b)
 (x
i 1
x
i 1
(d )
n
n
i 1
i 1
 y i )   x i  y i
n 1
n
( c)
i
i
 x n 1   x i
i 1
n
n
n
i 1
i 1
i 1
 x i yi   x i  yi
4.
Often, in economics the components of the summation operator are left off the
equation. This generally occurs when there is no ambiguous start and stop to the
summation. Therefore, often you will see the summation operator written as
 x i or  x i . Further, the components of the summation are sometimes written
i
on the side instead of underneath the summation operator ix i or

n
i
xi .
Answers
1.
(a )
2x 1  2x 2  2x 3
( b)
a 1 ( x 1  1)  a 2 ( x 2  1)  a 3 ( x 3  1)
(c)
b1 x 1  b 2 x 2  b 3 x 3
(a )
x
2.
5
i 1
3
( b)
i
1
x
i 0
x0
i
2
(c)
 a (x
i 1
i
3
(d )
a x
i 1
3.
i
i
 1)
i 1
i
Expanding each expression using n=3 and using simple algebra will show the
properties holds.
3
(a )
ax 1  ax 2  ax 3  a ( x 1  x 2  x 3 )  a  x i
i 1
This property shows a constant can be factored out of the summation notation.
( b)
This property shows the summation operator can be used in each additive
component of an equation.
4
( c)
x 1  x 2  x 3  x 31   x i
i 1
(d )
3
3
i 1
i 1
( x 1  y1 )  ( x 2  y 2 )  ( x 3  y 3 )  ( x 1  x 2  x 3 )  ( y1  y 2  y 3 )   x i   y i
x 1 y1  x 2 y 2  x 3 y 3  ( x 1  x 2  x 3 ) ( y 1  y 2  y 3 )
This property shows the summation operator cannot be moved through
multiplication (division) in an equation.