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(单词翻译:双击或拖选)
by Jason Marshall
In the last few articles, we’ve talked about fractions and percentages, and soon enough we’ll see that these ideas naturally lead us into the world of decimal numbers. But before we head down that path, let’s take a quick detour1 to talk about what I consider to be a rather beautiful area of math—sequences and series. Today, we’ll discuss a particular type of sequence known as an arithmetic sequence. Then, in the weeks to come, we’ll take a look at geometric sequences, the famous Fibonacci sequence, and some truly fascinating mathematical series.
But before we get to any of that, the podcast edition of this tip was sponsored by Go To Meeting. Save time and money by hosting your meetings online. Visit GoToMeeting.com/podcast and sign up for a free 45 day trial of their web conferencing solution.
What is a Mathematical Sequence?
In both math and English, a “sequence” refers to a group of things arranged in some particular order. Outside of math, the things being arranged could be anything—perhaps the sequence of steps in baking a pie. But in math, the things being arranged are usually—no surprise here—numbers.
One example of a sequence is the list of numbers:
1, 2, 3.
Or, as an example of an entirely2 different sequence:
3, 2, 1.
Yes, both of these sequences have the same elements or members (1, 2, and 3), but they’re arranged in a different order—so they are, in fact, entirely different three-element long sequences. Of course, sequences don’t always have to have three elements—they can have any number of elements. For example:
2, 3, 5, 7, 11
is the sequence containing the first five prime numbers (those are natural numbers only divisible by themselves and 1). But why stop at five?—sequences can even be infinite! But how do you write something that’s infinitely3 long?
How to Write Mathematical Sequences
Okay, let’s briefly4 talk about the notation5 used to write sequences—including those that are infinitely long. First, the elements of a sequence are usually written out in a row, with each element separated by a comma. Sometimes the elements are grouped together inside parenthesis6 like
( 2, 3, 5, 7, 11 ),
but not always.
How to Write Mathematical Sequences That Are Infinitely Long
If a sequence has infinitely many elements, we indicate that by writing ellipses7 at the end of the sequence if it extends out indefinitely in the positive direction, or at the beginning of the sequence if it extends out indefinitely in the negative direction. For example, the sequence of positive integers can be written
1, 2, 3, 4, 5, …
The “…” indicates the sequence continues forever in the positive direction. The sequence of negative integers can be written
…, -5, -4, -3, -2, -1.
Here, the “…” indicates the sequence continues forever in the negative direction. Putting these two together, the sequence of all integers can therefore be written
…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...
What are Arithmetic Sequences?
Now let’s talk about a specific type of mathematical sequence: the arithmetic sequence. I know it sounds complicated, but it’s really pretty simple. An arithmetic sequence is a sequence of numbers where the difference between any two successive elements is always the same constant value. For example, the sequence of years since the start of the new millennium8 is an arithmetic sequence:
2001, 2002, …, 2009, 2010.
Why is this an arithmetic sequence? Because the difference between all successive elements is always the same—2002 – 2001 = 1, 2010 – 2009 = 1—the difference is always 1.
Notice I’ve used ellipses here in the middle of the sequence. What does that mean? Well, ellipses are used like this to represent missing elements—in this case: 2003, 2004, and so on, up to 2008. I could have written them all out explicitly9, but using ellipses saves some writing.
What are Even and Odd Numbers?
The difference between successive elements in an arithmetic sequence doesn’t have to be 1—in fact, it can be anything. There are two famous arithmetic sequences you’re already familiar with whose successive members have differences of 2: the even and odd positive integers. Positive even integers begin at 2 and increase in steps of 2:
2, 4, 6, 8, 10, …
whereas positive odd integers begin at 1 and increase in steps of 2
1, 3, 5, 7, 9, …
Properties of Even and Odd Numbers
The members of these two sequences have some interesting properties. Whenever you add two even integers together, or two odd integers together, the answer is always an even number. For example, 2 + 6 = 8, 1 + 5 = 6, or 11 + 17 = 28—always even! However, whenever you add one even and one odd integer together, the answer is always odd. For example: 8 + 3 = 11 or 22 + 9 = 31—always odd!
Here’s a quick and dirty tip based upon this that can help you check your work: When you’re adding up numbers, you can use what’s called the “parity” of the numbers (that is, whether the numbers—or terms—you’re adding are even or odd), to make sure you have the right answer! If there are an even number of odd terms in your addition problem, the final answer must be even. However, if there are an odd number of odd terms in your problem, the final answer must be odd. For example, say you’re adding 23 + 6 + 79. Before even starting to add the numbers, I already know the answer must be even because there are an even number of odd terms (two, in this case: 23 and 79). This trick can be handy in everyday life, but it really shines when used on tests like the SAT or GRE to easily eliminate some of those multiple choices!
Brain-Teaser Problem
Next time, we’ll continue our tour of mathematical sequences with a look at geometric sequences. Until then, here’s a problem dealing10 with arithmetic sequences for you to contemplate11:
Can you think of a more efficient way to fully12 define an arithmetic sequence other than simply writing out all its elements?
This one is a bit tricky13. So think about it, and then look for the answer in this week’s Math Dude Video Extra! episode on YouTube and Facebook.
Wrap Up
Thanks again to our sponsor this week, Go To Meeting. Visit GoToMeeting.com/podcast and sign up for a free 45 day trial of their online conferencing service.
Please email your math questions and comments to。。。。。。。get updates about the show and my day-to-day musings about math, science, and life in general by following me on Twitter, and join our growing community of social networking math fans by becoming a fan of the Math Dude on Facebook—it’s a great place to ask questions and chat with other math enthusiasts14.
If you like what you’ve read and have a few minutes to spare, I’d greatly appreciate your review on iTunes. And while you’re there, please subscribe15 to the podcast to ensure you’ll never miss a new Math Dude episode.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
收听单词发音 |
1
detour
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| n.绕行的路，迂回路；v.迂回，绕道 | |
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entirely
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| ad.全部地，完整地；完全地，彻底地 | |
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infinitely
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| adv.无限地，无穷地 | |
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briefly
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| adv.简单地，简短地 | |
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notation
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| n.记号法，表示法，注释；[计算机]记法 | |
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parenthesis
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| n.圆括号，插入语，插曲，间歇，停歇 | |
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7
ellipses
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| n.椭园，省略号；椭圆( ellipse的名词复数 )；（语法结构上的）省略( ellipsis的名词复数 ) | |
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millennium
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| n.一千年，千禧年；太平盛世 | |
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9
explicitly
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| ad.明确地，显然地 | |
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10
dealing
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| n.经商方法，待人态度 | |
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11
contemplate
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| vt.盘算，计议；周密考虑；注视，凝视 | |
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12
fully
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| adv.完全地，全部地，彻底地；充分地 | |
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13
tricky
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| adj.狡猾的，奸诈的；(工作等)棘手的，微妙的 | |
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14
enthusiasts
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| n.热心人，热衷者( enthusiast的名词复数 ) | |
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15
subscribe
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| vi.(to)订阅，订购；同意；vt.捐助，赞助 | |
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