Sequences and Series
It's All Starting To Add Up
If sequences are the regular season, series are the Super Bowl. You have to be successful during the football season to get to the big game. Series build on sequences the same way. A series is nothing more than the addition of the terms of a sequence. It just has some fancy-pants notation. If we were comparing math and baseball, series would be the World Series. (No relation.)
Series will typically be written like so:
Trust us when we say this is way more intimating looking than it actually is. The big guy there, ∑, is the Greek letter sigma. It means sum. This type of notation is technically called either sigma notation or summation notation. To-MAY-to, to-MAH-to.
The number written below sigma tells you where your terms will start. This is typically a 1 or a good ol' goose egg. The one up top will be where your terms finish. This top number can be whatever you'd like as well but infinity is by far the greatest. Side note: these can also be written like this: .
Find the sum of .
Solution: All this problem is asking us to do is to add the terms up following the rule n2 starting with n = 1 and ending with n = 5. Listing the terms out we get:
1 + 4 + 9 + 16 + 25 = Put that calculator away. What is this? Amateur hour?
1 + 4 + 9 + 16 + 25 = 54
So we could write that .
The sum of . Solve for x.
Solution: This is a little tougher, but we're just being asked to find the term number the series goes up to. The problem creator is likely just making sure we really understand how a series works.
All we need to do is start at n = 1 and add terms until we get to 16. Cake.
1 + 3 = 4
(Definitely not 16.)
1 + 3 + 5 = 9
1 + 3 + 5 + 7= 16
We added 4 terms.
x = 4.
Other neat-o math nerd tricks include multiplying sums by constant multiples. is the same as and is equal to 54 × 3, which is otherwise known as 162.
This is a basic concept that could get you out of a sticky situation or two. Like if a scary clown were to ever ask us to tell him what was without a calculator or else. We don't want to find out what "else" means.
You have been captured by a mathematically obsessed clown who would like to know the sum of . He does not have a calculator. He makes a deal with you that if you tell him, he'll let you go.
Solution: To start, we are going to use this constant multiple concept from before to rewrite the series.
That means that now looks like .
This lets us deal with the sum first and the multiplication later.
Forget that pesky 17 for now and just find . Here goes…
[3(1) – 1] + [3(2) – 1] + [3(3) – 1] + [3(4) – 1] + [3(5) – 1] + [3(6) – 1] =
That step above is just the first term plus the second plus the third…up to six.
[3 – 1] + [6 – 1] + [9 – 1] + [12 – 1] + [15 – 1] + [18 – 1] =
2 + 5 + 8 + 11 + 14 + 17 =
Bringing 17 back into the sweet, sweet clown problem gives us…
17 × 57 = 969
See. Useful. We hate to say it, but we told you so.