MCR3UW
Date: __________________
We will now look at ways to add the terms of a sequence. Adding the terms of a sequence gives us a series.
u1 , u2 , u3 , u4 , u5 ,..., un is a sequence.
u1  u2  u3  u4  u5 ,..., un is a series.
The Greek letter sigma (  ) is used to represent the sums of values in mathematics.
n
u
i 1
i
This means the sum of the first n terms of a sequence. And is read as ‘the sum of all the terms ui from i  1 to
i  n ’.
Consider the arithmetic sequence 2, 6, 10, 14,… has a first term of 2, and a common difference of 4. A
general rule for the nth term of this sequence is u n  4n  2 .
Therefore the sum of the first 5 terms of this sequence can be written as:
This means that ‘the sum of all the terms of 4n - 2 from n = 1 to n = 5’.
Let’s calculate the sum below:
5
 4n  2 =
n 1
4
Ex 1
Write the expression
x
x 1
2
 2 as a sum of terms, and then calculate that sum.
6
Ex 2
Evaluate the expression
2
a
a 3
Ex 3
Write the series 2 + 10 + 50 + 250 + 1250 + 6250 using sigma notation.
Arithmetic Series
The sum of the terms in an arithmetic sequence is called an arithmetic series.
When a series has only a few elements (like above), calculating the sum is not a difficult task. However if a
series has 50, or 75, or 100 terms, it would be very time-consuming to add all these terms. It will be helpful
to find a rule, or formula for evaluating arithmetic series.
S n denotes the sum of the first n terms of a series. For a series with n terms,
Sn  u1  u2  u3  u4  u5 ,..., un
For an arithmetic series, this would be:
Another way to look at this (by reversing the order, and starting with the biggest term would be:
If we then add these two equations together vertically (like in elimination with a system) we get:
We can simplify this to:
And when we divide both sides by 2 we get:
If we substitute un  u1  (n  1)d , we get:
In general, you can find the sum of the first n terms of an arithmetic series using the formula:
Sn 
Ex 4
Ex 5
n
n
 u1  un  or Sn   2u1  (n  1)d 
2
2
Calculate the sum of the first 15 terms of the series 28 + 22 + 16 + …
Find the number of terms in the series 14 + 15.5 + 17 + 18.5 + … + 50. and then find the sum of the
terms.
Ex 6
Write an expression for S n , the sum of the first n terms of the series:
70 + 65 + 60 + … Then find the value for n, which Sn  0