This tells us that it is possible to describe the sequence as a second degree polynomial but it does not give us any information about how. If we look at the difference between the five initial numbers we find that they are 3 5 7 9 and, as you can see, the differences between these numbers are 2. 2 5 10 17 26… is an example of such a sequence. If it turns out that the difference between the differences is constant it means that the sequence can be described using a second degree polynomial. If neither quotient nor difference is constant it might be a good idea to look at the difference between the differences. This sequence can be described using the exponential formula a n = 2 n. 2 4 8 16… is an example of a geometric progression that starts with 2 and is doubled for each position in the sequence. In a geometric progression the quotient between one number and the next is always the same. This sequence can be described using the linear formula a n = 3 n − 2. 1 4 7 10 13… is an example of an arithmetic progression that starts with 1 and increases by 3 for each position in the sequence. Well for an arithmeticĪmount regardless of what our index is.In an arithmetic progression the difference between one number and the next is always the same. That we're adding based on what our index is. So this looks close,īut notice here we're changing the amount
#Solving sequences plus#
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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