Fundamental definitions and laws Natural numbers In a collection or set of objects or elementsthe act of determining the number of objects present is called counting. The numbers thus obtained are called the counting numbers or natural numbers 1, 2, 3, …. For an empty set, no object is present, and the count yields the number 0, which, appended to the natural numbers, produces what are known as the whole numbers.
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A sequence is nothing more than a list of numbers written in a specific order. The list may or may not have an infinite number of terms in them although we will be dealing exclusively with infinite sequences in this class.
In the notation above we need to be very careful with the subscripts. This is an easy mistake to make when you first start dealing with this kind of thing. There is a variety of ways of denoting a sequence. Each of the following are equivalent ways of denoting a sequence. A couple of notes are now in order about these notations.
First, note the difference between the second and third notations above.
If the starting point is not important or is implied in some way by the problem it is often not written down as we did in the third notation. A sequence will start where ever it needs to start.
Example 1 Write down the first few terms of each of the following sequences. Sequences of this kind are sometimes called alternating sequences. However, it does tell us what each term should be.
Before delving further into this idea however we need to get a couple more ideas out of the way. It is the same notation we used when we talked about the limit of a function. Using the ideas that we developed for limits of functions we can write down the following working definition for limits of sequences.
The working definitions of the various sequence limits are nice in that they help us to visualize what the limit actually is. Just like with limits of functions however, there is also a precise definition for each of these limits. Now that we have the definitions of the limit of sequences out of the way we have a bit of terminology that we need to look at.
So just how do we find the limits of sequences? Most limits of most sequences can be found using one of the following theorems. We will more often just treat the limit as if it were a limit of a function and take the limit as we always did back in Calculus I when we were taking the limits of functions.
So, now that we know that taking the limit of a sequence is nearly identical to taking the limit of a function we also know that all the properties from the limits of functions will also hold.
Next, just as we had a Squeeze Theorem for function limits we also have one for sequences and it is pretty much identical to the function limit version.Sal introduces arithmetic sequences and their main features, the initial term and the common difference. He gives various examples of such sequences, defined explicitly and recursively.
Each number in the sequence is called a term (or sometimes "element" or "member"), read Sequences and Series for more details. Arithmetic Sequence In an Arithmetic Sequence the difference between one term and the next is a constant. Review sequences and then dive into arithmetic and geometric series.
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Use arithmetic sequence. Algebra > Sequences and Series > Arithmetic Sequences. Page 1 of 6. Arithmetic Sequences. These are arithmetic sequences: Gauss's Problem and Arithmetic Series. Geometric Sequences.
23) a. Aug 06, · Arithmetic sequences are sequences of numbers in which any two consecutive numbers in the sequence are the same distance apart.
For example: 3, 1, 5, 9, 13, 17, 21, 25, 29, 33, .. is an arithmetic sequence.