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 now when you hear the word pattern you might think of a shirt yep and i'll bet you wish you had some fine threads like these don't you oh actually i'm good with this shirt thanks i get it not everyone can pull off a look this rad that's true of course 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 well because i like bunnies well 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 147147 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 add the sequence 1 2 3 three two one the set of numbers in that sequence is just one two three 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 three four five six 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 six 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 2. 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 six elements this sequence is non-repeating and finite because it also has just six elements this sequence is repeating an infinite and this sequence is non-repeating and infinite makes 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 zero and one 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 one but then skip every other number you'd end up with the sequence one three five seven nine 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 two or suppose you start counting at the number two 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 can make other sequences by skip counting by different amounts like you could start with 0 and skip every two 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 first element to the second by adding two one plus two equals three and you could get from the fourth element to the fifth by adding two seven plus two equals nine 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 plus 2 equals 13. all four arithmetic operations can be used as rules for generating sequences you've already seen how addition rules 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 one yep you get a sequence that counts down or decreases five four three two one lift off the rule for this simple countdown sequence is to start with five and then subtract one 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 going to 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 times 2 which is 2. the next would be 2 times 2 which is 4 and the next would be 4 times 2 which is 8. then the next would be 8 times 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 to each time we got up to 13 by the seventh element but when we multiplied by two each time we got up to 64 by the seventh 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 and 34 then 32 and so on notice how the sequence that's based on the division rule gets smaller much faster than the sequence 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 going to end this video with some tips that 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 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 a 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 four and eight is four then pick any other two adjacent numbers and do the same thing i'm going to pick the last two twenty minus sixteen is also four 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 four 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 six divided by two we get three and if we take the next two adjacent elements and divide them like this eighteen divided by six we also get three 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 multiplied by three 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 bit 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 the sequence's rule involves 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 the sequence is rule involves 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 re-watch 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 you next time learn more at mathantics.com mr whisker says to like and subscribe