Intro to sequences. A sequence is an ordered list of numbers. So these are some examples of some sequences. All of these are just lists of numbers and each one is following some kind of pattern. Over the course of these few videos, we'll talk about all of the sequences. Read full transcript
Sequences are infinite. For the purpose of the test, every sequence follows some relatively easy pattern. Now, in real mathematics, of course, there are incredibly hard sequences, incredibly tricky sequences. You're not going to deal with that on the test.
Everything on the test will follow a nice easy pattern. Sometimes it will be the job of the test-taker to discern the pattern. More often, the question will present the pattern in some form and ask something else about the sequence, the value of a later term, or the sum or difference of two terms, something along those lines. Let's talk about the notation for sequences.
We represent the sequence as a whole by an individual letter, usually a lower case letter, and the individual number in the sequence, the, the order in the list, by a numerical subscript. So, for example, if I say A sub 5 equals 28, that means for some sequence, the fifth term, the fifth number on the list, equals 28. So 5 is the position on the list.
I go down to the fifth number on the list and the fifth number on the list is 28. We can also use the subscript notation to denote a general term of the sequence, s sub n. Here, n stands for any natural number. So n could be 1, 2, 3, 4, etc. One way to specify an entire sequence all at once would be to give an algebraic formula for s sub n.
For example, the equation r sub n equals n times n plus 2 defines a complete infinite sequence. For any natural number, we can plug in that number and get a number on the list. And so, this gives us an infinite list. We could find, say, the third number on the list simply by plugging in n equals 3, plugging that into the formula.
This subscript variable is also known as the index, which is a way to say the place on the list. So n is the index. Let's think more about this sequence, r sub n equals n times n plus 2. For practice, let's find the first few terms. So when n equals 1, we get 1 times 3, which is 3.
That's the first number on the list. When n equals 2, we get 2 times 4. That's 8, the second number on the list. For n equals 3, we get 3 plus 5. That's 15, the third number on the list. For n equals 4, we get 4 times 6, that's 24.
For n equals 5, we get 5 times 7, 35. And so, the sequence begins with those numbers. Those are the first five numbers on the list, 3, 8, 15, 24, 35. Now, it would be very hard if you were given that sequence to discern the pattern. The test is not going to expect you to do that, but they might give you the formula and ask you to find things, find any of these numbers, for example.
Notice that if a formula is given, it makes it very easy to jump ahead to any term we want. So, for example, the test could give us that formula and ask us for the 48th term on the list. And, of course, we could just jump right there just by plugging in n equals 48. We get 48 times 50, then use the doubling and halving trick, so half of 48 is 24.
We get 24 times 100, so that's 2,400. That's the 48th number on the list. Here's a practice question. Pause the video and then we'll talk about this. So the test likes these algebraic sequences that are defined in terms of fractions of some kind.
So here, we get an algebraically defined sequence. We want to find the difference of two terms. So first, we have to plug in n equals 10 and n equals 6. For n equals 10, we get a sub 10 equals one-twelfth. For n equals 6, we get a sub 6 equals one-eighth. And now we're gonna subtract these two, one-twelfth minus one-eighth.
This is two-twenty-fourths minus three-twenty-fourths. We actually get a negative number, negative one-twenty-fourth. And that's the answer. Some basic patterns of which to be aware. A sub n equals n, that's just the sequence of all positive integers, so just the counting numbers, 1, 2, 3, 4, 5, 6, etc.
That is the simplest of all possible sequences. A sub n equals 2n minus 1 is the sequence of all positive odd numbers. So very interesting. And if we just had add a sub n equals 2n, that would be the sequence of all positive, positive even numbers.
A sub n equals 7n is the sequence of all positive multiples of 7. And similarly, if we had any factor times n, it would be all the multiples of that particular factor. A sub n equals n squared is the sequence of all positive perfect squares. A sub n equals 3 to the n is the sequence of all the powers of 3. These are very simple patterns, but may be used as building blocks in more complicated sequences.
So here's a practice problem. Pause the video and then we'll talk about this. Okay. Which of the following could be true of at least some of the terms on the sequence defined by that algebraic formula?
So, could they be divisible by 2, divisible by 3, divisible by 5? Well, first of all, let's just notice if we plug in n equals 1, the simplest thing, the first number on the list, what we get is 5. So certainly, it's possible for some of the numbers to be divisible by 5. So that is possible. Three is possible.
Now, if we plug in n equals 2, then we get 21, which is divisible by 3, so it is possible that some of the numbers on the list are divisible by 3. Okay, so far so good. So two and three are possible. Now we get to one. Now notice, think about this, 2n.
If n is an integer, then 2n has to be an even integer. 2n minus 1 has to be odd and 2n plus 3 has to be odd. So, b sub n is a product of odd times odd. So, every number in this sequence is an odd number. So none of them are divisible by 2. So it turns out it is not possible for any number on that list to be divisible by 2 because they're all odd numbers.
So one is not possible, two and three are possible, and that gives us an answer of D. In summary, a sequence is an ordered list of numbers. In the notation a sub n, n is the index, that is, the place on the list. An entire infinite sequence can be specified simply by giving an algebraic formula for a sub n in terms of n.