Finding the Sum of a Series Using Sigma Notation

In math a sequence is a function that has a set of natural numbers as its domain. Natural numbers are all positive integers above zero, not including zero.

Take the following sequence for example. "n" is used for the variable as a reminder that this function is a sequence and n must be a natural number.
sigma1
If you want to find the sum of all values from 3 through 7 for this sequence, you could do it the algebraic way by plugging in 3 for n, then 4 for n, then 5 for n, then 6 for n, then 7 for n, then add up all of those terms, and you would find the sum.
It would look like this.
sigma2

If you added up 25, 30, 35, 40, 45 and 50, you would find the sum of all the values between n=3 and n= 7, is 175.
The pattern of this sequence, appears to be increasing arithmetically (by addition) by 5 units.  To confirm that is the pattern called the common difference, use the common difference formula:

sigma3


So you could plug in any term from this sequence that has a term before it (so not the first one) and then subtract the term before it from it to discover the common difference. If it is really an arithmetic sequence then the common difference will be the same for each pair of terms in the sequence that meet the definition of the formula.
Such as
45-40=5
40-35=5
The common difference of this arithmetic sequence is 5.

You could write the exact same problem using summation notation. To solve the problem using Sigma notation, you must be familiar with the Summation Properties and Rules.

sigma4

First you would use property 3 to separate the terms of the function (called a sequence) into two different groups like this.
sigma5

 

Then you would use property 2 to separate the variable from the constant in the first term like so.

 sigma6

For the first term in this expression, you must use rule 1 to replace sigma7with sigma8

 

sigma9
Noticing the sigma notation model, n in this function would be 7, therefore plug in 7 into summation rule 1 for the first term.
sigma10

This first term in the expression simplifies to:
 sigma11

From here, you multiply 28 by 5 which equals 140 and the first term in the expression further simplifies to 140.

For the second term in the expression you will use the property 1 which is the constant property.

sigma12

Using summation property 1 on the second term you will multiply 7 by 10 which equals 70. And the second term further simplifies to 70.

Your expression now looks like

140 + 70  = 210

However that is not the solution. There is still one more step. Because we want to find the sum of values between 3 and 7 we must subtract the
sigma13
and
sigma14
values.

sigma15

Therefore subtracting 35 from 210 gives the correct answer of 175.

 


by Andi Boggs

Posted: May 1st, 2015

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