First we must write an equation for the pattern in the series. We can see that this is an arithmetic pattern because the same number (4) is being added each time. 

The formula to write the rule or equation for an arithmetic pattern is: 

Where a1 is the first value in the sequence and d is the common difference, n is left as a variable. 

When we plug in the values we are given we get: 

Which simplifies to become: 

Which further simplifies to become: 

Now we must put this in the sigma or summation notation. 

We know that summation notation always involves the greek letter sigma 

The number on the bottom is the first number that is plugged in, first input or first domain number - in this case it is 1. 

The number on top is the last number that is plugged in, the last input or the number of numbers in the series - there are five numbers in this series, so it ends with 5. 

So the final summation will look like this: 

Evaluate each summation term first.

The bottom number is the first term of the summation. Plug the number into the expression. Repeat the summation for every whole number until the summation reaches the top number.

Evaluate the next summation.

Subtract this value from the value of the first summation.

The answer is:  

To evaluate this summation, first ignore the outside negative sign and substitute the bottom value into the quantity inside the summation.   Reiterate for every integer after zero until we reach the top integer.

Simplify.

Therefore:

The answer is .

The summation will loop from  to .

Rewrite the summation.

The answer is:  

In order to evaluate this summation, expand the terms.  Start by plugging in zero into , and repeat until we reach to two.

Simplify the terms by order of operations.

Evaluate the terms inside the parentheses first.

The answer is:  

In order to solve the summation, expand the terms of the binomial.  Substitute two first, and add the quantities of each term for each integer repeating until the top integer is reached.

The answer is:  

Write out the terms of the sigma notation.  This will repeat for the second to the fifth term.  There is no variable to substitute, which means that after every iteration, the value that is summed is two.

The answer is:  

In order to evaluate the summation, substitute the bottom number into the quantity, and repeat for every integer after the eight and up to the top number, ten.

Evaluate each term and simplify.

The answer is:  

This summation will repeat three times, for .

Substitute the first term into the parentheses, repeat, and sum the process for the next three terms.

Simplify each term.

The answer is:  

Expand the summation sign.  Start with zero as the first index, then one, and finally two.  Since two is the top digit, the summation will stop.

Simplify these terms.

The answer is: