Let's use examples to derive some summation properties. First recall that where a_k stands for the k^th term of the sequence.

1. Sum of a constant:

Example 1: Evaluate and and use your discovery to evaluate:

 

 

Fill in the general property:

2. A constant multiplied by each term of the sequence:

Example 2: Suppose . Evaluate

 

 

Fill in the general property:

Note that although we used a finite sum in this example, the property would also work for infinite sums:

 

 

3. Break up a summation:

Example 3: Given: and and , evaluate .

 

 

Fill in the missing parts of the general property:

where m < n and m, n are positive integers.

 

 

4. Adjust the indices of a summation:

Example 4: Given that: , evaluate

 

 

 

Fill in the missing parts of the general property:

 

 

5. Break up the argument of the summation when there is more than one term:

Example 5: Evaluate if we know:

 

 

 

In general:

 

 

 

6. Pairing of terms:

In some special cases, you can pair off terms and use this pairing to find the sum. (This is also an arithmetic series.):

Find a shortcut to evaluating:

 

 

In general:

Of course you could make up all kinds of problems that can use one or several of these properties. See the summation problem set.

Warning: . Can you figure out why?