Here’s how it works: type in the known numbers in order separated by commas. The first example, 3, 5, 2, 7, 10 gave me sequence A122805; the preceding number is 4. The third, 6, 10, 12, 8, 11 gave me sequence A057368; the preceding number is 8.

As the first responder explained, these exercises are a little silly since one can create a polynomial that has whatever roots you want. There are, however, certain integer sequences that do come up in math.

The HW does seem excessive and inadequately presented.

http://www.research.att.com/~njas/sequences/index.html

Beyond that, there are truthfully an infinite number of possible right answers. What your son is being asked to do in part here is exercise induction – looking at patterns and extrapolating a rule to fit them. I don’t know how your son’s teacher would respond to this line of reasoning, but any pattern your son sees is a valid answer to the question. To paraphrase Robert Heinlein – deductive logic is always right, but never tells you anything you didn’t already know. Inductive logic sometimes gets it wrong but can give you new truths. I guess my point here is don’t be too upset if you can’t get “the right answer” to these. Pretty much anything you put down can be argued as “a right answer”.

As to the 20 pages – well, it seems excessive to me, but it depends on how much work those 20 pages actually represent, how much time they were given in class to work on them, and other factors like that. If you’re concerned about it, I’d suggest you talk to his teacher. And please – talk to the teacher before you go over her head. I’ve had that happen more than once, and it’s truly irritating.

Edit: If that’s all he was given, then I’m afraid I have no idea where to go from here except what I already said. The question is a crap-shoot, so roll the dice.

My thumbrule for middle school students is about half a period of homework per night. If the period is 45 minutes long, then 20-25 minutes of homework. It appears this goes well beyond that, so I’d be on your wife’s side. Again, I’d urge you to talk to the teacher about it – if for no other reason than to air your concerns.

Sorry – wish I could help.