Working with Sequences
Date: 11/14/98 at 09:16:12 From: Kathy Kwan Subject: Sequences I have tried to do the following questions twice, but I still couldn't figure them out. Would you please show me how to do them? 1) Give the next two terms of the sequence: 1, 1, 2, 4, 3, 9, 4, ... 2) Write down the first four terms and the seventh term of the sequence for which the nth term is given. a) 2n - 1 b) n - 4
Date: 11/19/98 at 21:20:46 From: Doctor Anderson Subject: Re: Sequences Hi, Kathy. These can be tricky, especially the first one, where you have to find a pattern. You can think about it for hours, trying a million different patterns, and never get the right one. I don't know about you, but my first instinct on question (1) is to figure out what to add to or subtract from each term to get the next term. This is often a good method, and we can sometimes find the answer easily by writing out this sequence: 1-1, 2-1, 4-2, 3-4, 9-3, 4-9, ... which is the same as: 0, 1, 2, -1, 6, -5,... So to get the second term, add 0 to the first, to get the third, add 1 to the second, to get the fourth, add 2 to the third, to get the fifth, add (-1) to the fourth, etc. Well, I don't see a nice pattern, do you? So although this often works, it doesn't seem to here. Let's think of another way to look at it. I can't really help you here without telling you the right way to go. I can't give you the actual answer, but here is the right direction on this one. Pair up terms that are next to each other. Do it like this: (1,1), (2,4), (3,9), and you can't make the next pair yet, call it (4,x). Look at these, especially (2,4) and (3,9), for a while. What kind of relation does the second number have with the first, in each pair? This should help you find x, once you see the pattern. Now, that gives you the next term, so what about the one after that? Well, look at the sequence of first terms of each pair. This has a really simple pattern. So far we have (1,1), (2,4), (3,9), and (4,x), so let's call the next pair (y,z). I hope you can find y without too much trouble, and you don't even need to find z (the question doesn't ask for it). Now, for question (2). I will solve an example like yours, all the way through. Let's use the sequence with the nth term being 3n+2. What is another way of saying the first term? It is the term where n = 1. Well, if n = 1, then 3n + 2 = 3(1) + 2 = 5. So we can make a simple chart following this procedure: n 3n+2 ------------------ 1 3(1) + 2 = 5 2 3(2) + 2 = 8 3 3(3) + 2 = 11 4 3(4) + 2 = 14 7 3(7) + 2 = 23 From here, you should be able to find the first 4 terms and the 7th term of your problems. Good luck, and if you get stuck, feel free to ask for more help. - Doctor Anderson, The Math Forum http://mathforum.org/dr.math/
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