Hi, I’m Rob. Welcome to Math Antics!
By now you probably know that math involves a lot of calculations using arithmetic.
But math is about more than just calculations.
In fact, one important type of math that sometimes gets overlooked involves number patterns.
When you hear the word “pattern”, you might think of a shirt.
Yep! And I’ll bet ya wish you had some fine threads like these, don’t ya?
Oh… Actually…I’m good with this shirt, thanks….
I get it… not everyone can pull off a look this rad.
That’s… that’s true.
The reason you might think of a shirt is because the word “pattern” often describes repeating images or objects.
Like if I show you this pattern, “dog, cat, bird, dog, cat, blank”.
What animal do you think should fill in the blank to complete the pattern?
A bunny!
Why would you think it was a bunny?
Because I like bunnies.
Well… It’s not a bunny. It’s a bird!
See how the pattern repeats?
“dog, cat bird, dog, cat, bird”.
Well Mr. Whiskers and I prefer the pattern,
“dog, cat, bunny, dog cat, bunny”.
Anyway, number patterns can be formed by repeating numbers like this:
1, 4, 7, 1, 4, 7
Notice how the order of the pattern really matters?
If you switch any of the numbers, it becomes a different pattern.
In math, when you have a set of numbers or elements where the order matters, it’s called a “Sequence”.
For example, the sequence 1, 2, 3 is different than the sequence 3, 2, 1
even though they each contain the same “set” of numbers.
And in math, the word “Set” refers to a group of numbers or elements
where the order doesn’t matter AND where any duplicates are left out.
For example, if you had the sequence 1, 2, 3, 3, 2, 1
the set of numbers in that sequence is just 1, 2, 3 even though each number occurred twice in the sequence.
Both sets and sequences use the same notation in math.
Each number or element is separated by a comma, and the whole group is put inside curly braces like this.
Some sequences of numbers repeat, like the sequence {0, 1, 0, 1, 0, 1}
but some don’t repeat, like the sequence {1, 2, 3, 4, 5, 6}.
But think about both of those sequences for a second…
Right now, each of them contains a limited or “finite” number of elements.
They each have 6.
But each of these sequences could be continued forever if we wanted to.
We could just keep repeating 0,1,0,1 forever
or we could just keep counting, 7, 8, 9, 10, forever too.
In other words, sequences can be finite OR they can be “infinite”.
If a sequence or set is finite, it means that you can say there are a specific number of elements in it,
like 6 or 20 or a million.
But when something is infinite, it means that no matter how much time you have,
you could never finish counting how many elements are in it.
You can’t give it a specific number so you just say it goes on forever.
Of course, we can’t actually write numbers forever on a piece of paper,
so we need to use a special notation for infinite sets or infinite sequences.
You just put three dots at the end of the list to show that it keeps on going forever like this.
The three dots are an abbreviation that means the sequence continues in the same way.
They can be used in the middle of a sequence to save writing.
Like this means the sequence of all counting numbers from 1 to 100.
But you can also use them at the end of a sequence to show that it goes on forever.
So this sequence is repeating and finite because it has just 6 elements.
This sequence is non-repeating and finite because it also has just 6 elements.
This sequence is repeating and infinite.
And this sequence is non-repeating and infinite. …make sense?
For these last two infinite sequences, what’s the set of numbers that each contains?
Well, the first keeps on repeating two numbers forever, so even though the sequence is infinite,
the set it uses is finite because it only contains 0 and 1.
But in the second infinite sequence, none of the elements are ever repeated,
so the set of numbers is exactly the same as the sequence itself. It’s also infinite.
Okay, so now you know that some number patterns are repeating and some aren’t.
You also know that some number patterns are finite and some are infinite.
We got our first non-repeating infinite sequence simply by counting.
Let’s see if we can think of some others that way too.
Suppose you start counting at the number 1 but then skip every other number.
You’d end up with the sequence {1, 3, 5, 7, 9,…} and so on.
In other words, you’d end up with the infinite, non-repeating sequence
that we call “odd numbers” because none divide evenly by 2.
Or, suppose you start counting at the number 2 instead, but still skip every other number.
You’d end up with the sequence {2, 4, 6, 8, 10,…} and so on.
That’s the infinite, non-repeating sequence of numbers we call “even numbers” because all divide evenly by 2.
And you could make other sequences by skip counting by different amounts.
Like you could start with 0 and skip every 2 numbers to get the sequence {0, 3, 6, 9, 12…} and so on.
If you think about it, counting (and skip counting) are really just ways of making a number sequence by following a “Rule”.
In the case of regular counting, that rule happens to be ‘Add 1’ to get each new number in the sequence.
And when you skip count every other number, the rule you’re following is ‘Add 2’ each time.
You can see that by looking at the sequence we called “odd numbers”.
You could get from the 1st element to the 2nd by adding 2 (1 + 2 = 3),
and you could get from the 4th element to the 5th by adding 2 (7 + 2 = 9)
In other words, if you know the rule that a particular sequence is based on,
you can use it to find any other number in the sequence.
If you want to know what number comes next in the sequence of odd numbers, just add 2 to the last element you know.
Like 11 + 2 = 13
All four arithmetic operations can be used as rules for generating sequences.
You’ve already seen how addition rues produce sequences that count up (or increase)
but what do you think you’d get if you based a sequence on a subtraction rule, like ‘subtract 1’?
Yep, you’d get a sequence that counts down (or decreases).
5, 4, 3, 2, 1, Liftoff!
The rule for this simple countdown sequence is to start with 5 and then subtract 1 each time.
Oh, and some of you who are familiar with negative numbers will realize that
this countdown sequence really doesn’t’ have to stop at zero.
It could continue on forever in the negative direction,
but we’re just gonna focus on positive numbers in this video.
Another example of a subtraction sequence is to start with 50 and use the rule ‘subtract 5’.
In that case you’d get {50,45,40,35, 30,…} and so on.
Each element in the sequence is 5 less than the one before it.
So it’s pretty easy to see how addition and subtraction can be the rule for a sequence…
but what about multiplication and division?
What number sequence would you get from the rule ‘multiply by 2’?
Well, if we start with 1 as the first element,
The next would be 1 x 2 which is 2.
The next would be 2 x 2 which is 4.
And the next would be 4 x 2 which is 8.
Then the next would be 8 x 2 which is 16, and so on.
Notice that the numbers in this sequence are getting big pretty fast.
That’s one of the clues that a sequence might be based on a multiplication rule.
When you keep multiplying a previous result by the same factor,
the values can grow much faster than if you just added a fixed amount each time.
You’ll see that if we compare the sequence we just made by multiplying by 2 each time,
with the sequence we previously made by adding 2 each time.
Even though both sequences start at the same number,
when we added 2 each time, we got up to 13 by the 7th element,
but when we multiplied by 2 each time we got up to 64 by the 7th element.
That’s quite a difference!
And it works in a similar way with division.
Suppose you’re asked to make a sequence by starting with 40 and then dividing by 2 each time.
The first number is 40.
The next is 40 divided by 2 which is 20.
The next is 20 divided by 2 which is 10.
The next is 10 divided by 2 which is 5.
The next is 5 divided by 2 which is 2.5
And we could keep on going, dividing by 2 forever to get smaller and smaller fractions
but we’ll stop there so we can compare that to the sequence you’d get if you start with 40 but subtract 2 each time.
In that case you’d get 40, then 38, then 36, then 34, then 32 and so on.
Notice how the sequence that’s based on the division rule gets smaller much faster
than the sequences that’s based on the subtraction rule.
…just like the sequence that’s based on multiplication got bigger much faster than the one based on addition.
That’s because when you keep adding or subtracting the same amount,
the sequence changes by a constant amount each step.
…just like going up or down a normal flight of stairs.
But if you multiply or divide each time,
the sequence changes by an increasing or decreasing amount each step.
That would be a tough set of stairs to climb!
In fact, there’s such a big difference in the way these types of sequences increase or decrease
that mathematicians have different names for them.
Sequences that are based on addition or subtraction rules are called “Arithmetic Sequences”
while sequences that are based on multiplication or division rules are called “Geometric Sequences”.
Those maybe aren’t the most intuitive names, but since they’ve been used for so long,
it’s important to know what people mean when they say them.
Okay, by now you’ve probably realized that there are LOTS of different kinds of number sequences and patterns in math
…far too many to cover in just one video.
So instead of trying to do that, we’re gonna end this video with some tips you can use
to figure out if a sequence is based on a simple rule involving addition, subtraction, multiplication or division.
When you’re given a sequence, first try to determine if it’s repeating or non-repeating.
For example, in this sequence, you can see that part of the sequence keeps repeating.
That means that you’d need to use the pattern to fill in any missing elements instead of a rule.
But if the sequence isn’t repeating, like this one,
the next thing you’d want to check is if the sequence is increasing or decreasing.
Not all sequences increase or decrease,
but increasing sequences could be based on an addition or multiplication rule,
while decreasing sequences could be based on a subtraction or division rule.
This sequence is increasing since each new element is bigger than the one before it,
but how can we tell if it’s based on an addition or a multiplication rule?
To do that, we need to look for either a “common difference” or a “common ratio” in the sequence.
Here’s what that means…
Start by picking any two adjacent numbers in the sequence and find the difference between them by subtracting.
For example, the difference between 4 and 8 is 4.
Then, pick any other two adjacent numbers and do the same thing.
I’m gonna pick these last two: 20 minus 16 is also 4.
Are the differences the same?
In this case yes!
That means that we have found what’s called a “common difference” for the sequence.
The common difference is a constant amount that’s either added or subtracted to each new element.
Since the common difference here is 4 and the sequence is increasing,
that means the rule for this sequence is probably ‘add 4’.
You can check to make sure all the other elements are following that rule just to be sure.
But what if we don’t find a common difference for a sequence? …like this one.
If we take the first two elements and subtract them, we get 4,
but if we take the next two elements and subtract them we get 12.
That means that there’s NOT a common difference for this sequence
so it’s not based on a simple addition or subtraction rule.
But, maybe we can find a common ratio instead. Let’s see what that means.
To find a common ratio, we also take two adjacent pairs of elements,
but instead of subtracting them, we divide them.
For example, if we take the first two elements and divide them like this (6 divided by 2) we get 3.
And if we take the next two adjacent elements and divide them like this (18 divided by 6) we also get 3.
Ah Ha! That’s what we call a common ratio.
And it means that this sequence is likely based on either a simple multiplication or division rule.
Since this is an increasing sequence, we know that the rule is probably ‘multiply by 3’.
Again, you can double check that on other pairs.
So even though not all sequences are based on simple arithmetic rules, checking for a common difference or a common ratio can help you identify the ones that are.
Alright, so now you know a little it about number sequences.
You know the difference between a sequence and a set.
You know that some sequences repeat while others don’t.
You know that some sequences are finite while others are infinite.
And you know that sequences can be based on arithmetic rules.
If a sequence’s rule involve adding or subtracting a constant amount each time,
that means you’ve got an “Arithmetic Sequence”
and you’ll be able to figure out that constant or “common difference” by subtracting pairs of adjacent numbers.
But if a sequence’s rule involve multiplying or dividing by the same factor each time,
that means that you’ve got a “Geometric Sequence”
and you’ll be able to identify its “common ratio” by dividing pairs of adjacent numbers.
We covered a lot in this video, so be sure to rewatch it later if it didn’t all sink in the first time.
And remember, the best way to get good at math is to practice what you’ve learned.
As always, thanks for watching Math Antics and I’ll see ya next time.
Learn more at www.mathantics.com
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