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SUMMATION NOTATION
Jump right to the exercises!
Summation notation gives a compact way to represent sums, when the terms exhibit some common pattern.
For example, the sum
1+
4+
9+
16+
25+
36+
49+
64+
81+
100
can be represented using summation notation as
∑
i=1
10
i2
or
∑
j=1
10
j2
or
∑
k=1
10
k2
or
∑
m=1
10
m2
or
∑
n=1
10
n2
.
Summation notation is also called sigma notation.
Some comments on
∑
i=1
10
i2
:
-
∑
i=1
10
i2
is read aloud as "the sum, as i goes from 1 to 10, of
i squared".
- i is called the index of summation.
The five most common letters to use for the index of summation are i, j,
k, m and n.
- The number 1 in
" i=1 " is called the lower limit
of summation.
This gives the starting value for i .
- The number 10
is the upper limit of summation.
This gives the ending value for i .
i starts with the lower limit and is incremented by one until
the upper limit is reached.
- The expression next to the summation symbol gives the pattern for each of the terms in the sum.
Here are some examples:
∑
j=-1
2
(
j3+1
)
=
(-1)3
+1
⏞
j=-1
+
03
+1
⏞
j=0
+
13
+1
⏞
j=1
+
23
+1
⏞
j=2
= 12
∑
k=3
7
xk
=
x3+
x4+
x5+
x6+
x7
∑
n=-2
3
5
=
5⏞
n=-2
+
5⏞
n=-1
+
5⏞
n=0
+
5⏞
n=1
+
5⏞
n=2
+
5⏞
n=3
=
6⋅5=30
PROPERTIES OF SUMS:
Let a and b be integers,
with a less than b .
Then,
∑
i=a
b
(
xi
+
yi
)
=
∑
i=a
b
xi
+
∑
i=a
b
yi
and
∑
i=a
b
(
xi
-
yi
)
=
∑
i=a
b
xi
-
∑
i=a
b
yi
Ideas:
(x1
+
y1)
+
(x2
+
y2)
=
(x1
+
x2)
+
(y1
+
y2)
(x1
-
y1)
+
(x2
-
y2)
=
(x1
+
x2)
-
(y1
+
y2)
So, you can split sums and differences apart!
∑
i=a
b
k
xi
=
k
∑
i=a
b
xi
Idea:
(kx1
+
kx2)
=
k(
x1
+
x2
)
So, you can slide constants out!
∑
i=a
b
k
=
(
b-a+1
)
k
Idea:
Consider this list of numbers: 7, 8, 9, 10, 11
How many numbers are listed here?
Answer: 5 numbers are listed.
The answer is NOT just 11 - 7! Instead, it's one more!
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.