It is defined as a sequence where every number has the same constant added to it. Arithmetic sequences are a special type of sequence. There are many special types of sequences. In this case, it would be, which can be anything:, ,, etc. The nth number is defined as the number n of any sequence. ![]() This number, at position 2, becomes the input for getting our next number at position 3. Because it is the first number in our list, it is located in position number 1. Using the previous example, we can break down each element. These rules can also be called functions because they take one number as input and give an output. The order is important, you can’t switch up the orderĪ rule is defined for how to get each successive numberĪs you can see, sequences have specific notation that we can use to define the rules they use. There are two special properties about sequences that go hand in hand. A sequence is a list of numbers that can be either finite or infinite. The most common type of pattern in math is called a sequence. Let’s take a look at some examples of patterns in math. In order to start discussing what ‘nth’ means, we should define a pattern.Ī pattern is any sort of trend in a list of numbers, objects and more. Well for an arithmeticĪmount regardless of what our index is.In calculus, many of the questions that you will encounter will ask you a question that relates to patterns in the function or numbers. That we're adding based on what our index is. So this looks close,īut notice here we're changing the amount Previous term plus whatever your index is. Or greater, a sub n is going to be equal to what? So a sub 2 is the previous It's going to infinity, with- we'll say our baseĬase- a sub 1 is equal to 1. So we could say, this isĮqual to a sub n, where n is starting at 1 and This, since we're trying to define our sequences? Let's say we wanted toĭefine it recursively. So this, first of all,Īrithmetic sequence. We're adding a differentĪmount every time. Giveaway that this is not an arithmetic sequence. Is is this one right over here an arithmetic sequence? Well, let's check it out. To the previous term plus d for n greater Wanted to the right the recursive way of defining anĪrithmetic sequence generally, you could say a subĮqual to a sub n minus 1. And in this case, k is negativeĥ, and in this case, k is 100. That's how much you'reĪdding by each time. So this is one way to defineĪn arithmetic sequence. Number, or decrementing by- times n minus 1. If you want toĭefine it explicitly, you could say a sub n isĮqual to some constant, which would essentiallyĬonstant plus some number that your incrementing. Wanted a generalizable way to spot or define anĪrithmetic sequence is going to be of the formĪ sub n- if we're talking about an infinite one-įrom n equals 1 to infinity. Than 1, for any index above 1, a sub n is equal to the ![]() One definition where we write it like this, or weĬould write a sub n, from n equals 1 to infinity. To define it explicitly, is equal to 100 plus Of- and we could just say a sub n, if we want Is the sequence a sub n, n going from 1 to infinity So this is indeed anĬlear, this is one, and this is one right over here. Is this one arithmetic? Well, we're going from 100. The arithmetic sequence that we have here. So either of theseĪre completely legitimate ways of defining And then each successive term,įor a sub 2 and greater- so I could say a sub n is equal We're going to add positiveĢ one less than the index that we're lookingĮxplicit definition of this arithmetic sequence. So for the secondįrom our base term, we added 2 three times. We could eitherĭefine it explicitly, we could write a sub n is equal With- and there's two ways we could define it. ![]() So this is clearly anĪrithmetic sequence. Then to go from negativeġ to 1, you had to add 2. These are arithmetic sequences? Well let's look at thisįirst one right over here. Term is a fixed amount larger than the previous one, which of So first, given thatĪn arithmetic sequence is one where each successive The index you're looking at, or as recursive definitions. ![]() And then just so thatĮither as explicit functions of the term you're looking for, Out which of these sequences are arithmetic sequences. Term is a fixed number larger than the term before it. Video is familiarize ourselves with a very commonĪrithmetic sequences.
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