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Finding Summations of Arithmetic Series

First, remember that ARITHMETIC means ADDITION, and SERIES is a SUM of a list of numbers. The SUMMATION sign, , is sigma from the Greek Alphabet. Anytime you see , think SUM!

New Vocabulary

  means SUM!

What does it mean?

5 n=-3

3n - 2

In order to solve this problem, we'll make a chart. One column of the chart will be n (our variable) and the other side will be 3n - 2 (the function). We simply fill in our n values to 3n - 2 in order to get the other value. For instance, we can plug in -1 for n. Hence, 3n - 2 = 3(-1) - 2 = -3 - 2 = -5. Notice that when n = -1, we have -5 under 3n - 2.

n    3n - 2 
-----------
-3    -11 
-2    -8 
-1    -5 ← See the -5 here?
0    -2 
1    1 
2    4
3    7 
4    10 
5    13 

Finally, we SUM up all the numbers on the right:
-11 + -8 + -5 + -2 + 1 + 4 + 7 + 10 + 13 = 9.

This is too much work!

What happens when we add -11 + 13? (we get 2)
-11 + -8 + -5 + -2 + 1 + 4 + 7 + 10 + 13 
 \____________________________________/

And what about -8 + 10? (2)
-11 + -8 + -5 + -2 + 1 + 4 + 7 + 10 + 13
       \_________________________/

And -5 and 7? (2) (see a pattern yet?)
-11 + -8 + -5 + -2 + 1 + 4 + 7 + 10 + 13
            \________________/

How about -2 and 4?
-11 + -8 + -5 + -2 + 1 + 4 + 7 + 10 + 13
                 \_______/

Since there are 9 values total, the total number of groupings is 4 ¹/₂ = ⁹/₂ = the number of values divided by 2!

And we know that the sum is ⁹/₂(-11 + 13)

In other words, the sum of the first n terms is:
S_n = ⁿ/₂(first term + last term)

Example #1

3 n=0

5 - 2(n-1) =

n        5 - 2(n-1)
-------------------
0        7
1        5
2        3
3        1

Example #2

229 n=-30

2n + 5 =

In this case, it would be easier to use the formula. So we'll begin by finding the first and last terms:
The first term (filling in n = -30) :
2(-30) + 5 = -60 + 5 = -55
The last term (filling in n = 229) :
2(229) + 5 = 458 + 5 = 463

Next, subtract the upper and lower bounds and add 1 to get the number of terms:
(229) - (-30) + 1 = 229 + 30 + 1 = 260.

Now, we'll fill in the formula:
S_n = ⁿ/₂(first term + last term)
S_n = ⁽²⁶⁰⁾/₂(-55 + 463) = (130)(408) = 53040

Aren't you glad we used the formula? I wouldn't want to make a chart with 260 entries!! (Warning: Only use the formula in the case of ARITHMETIC sequences!! )

Test Taking Strategies

For multiple choice tests, the best thing you can do is remember that means SUM. That way no matter what function you are given, you can always find the sum in a chart!

Review What You Learned

Example #3 (You try it!)

5 n=2

n2 =

Hint: This function is NOT arithmetic!!

Example #4 (You try it!)

25 n=2

-7 + 2(n-1) =