Awkward Sequence Problems

(A new question of the week)

Having just discussed some pattern or sequence problems that were poorly posed, let’s look at some recent questions about sequences, some of which are quite complicated, and others seem to be just wrong.

Two extreme patterns

Here is the initial question from Zehra:

Please make me understand how I complete these series:

1, 2, 5, 1, 3, 6, 2, 5, 8, 4 ………

10, 2, 4, 40, 8, 10, ……….

These are not, like some we looked at last time, too short to have sufficient information; but they look complex enough to require context. I wrote back:

Perhaps you have seen my posts on questions like this:

Pattern and Sequence Puzzles

Pattern and Sequence Puzzles Revisited

One important point made there is that problems like this are puzzles (or riddles): without a context limiting the type of pattern to expect, there is no certainty that there will be only one answer, and no routine method that will find it. We are really just guessing what sort of pattern would have interested the poser.

So, can you tell me the context of the questions? That will include the age or grade of the student for whom it was given, what they have been learning that it might be related to, and anything that was said about the problems.

Also, if you didn’t give the complete statement exactly as given to you, please do; as some of my examples show, it can be very important whether you are asked for a formula, or the next term, or something else. If there were other similar questions you have solved, it may help to see them for comparison.

I can suggest a way to find one possible answer to the first: Make the sequence of first differences and look for a pattern. The pattern I see is not a mathematical formula, but an alternation between three different patterns in the differences. (Compare the second example in my second post.)

I don’t yet see a pattern in the second, but I see reason to wonder if you might have made a small mistake in typing it. I’ll give it more thought when I hear back from you!

Zehra replied:

Well, it is for a student of Grade 5. Basically there is some pattern that we have to follow and complete the series like I am sharing some examples:

15, 18, 21, 24 ….

27, 30, 33, 36 ….

another:

10, 15, 10, 50, 10, 16 …..

10, 60, 10, 17, 10, 70 …..

I think these are two examples, in each case the first line being the given values, and the second being the next few terms that were asked for. The fact that only terms were requested clarifies that no formula is expected.

These examples are also useful in showing what sort of ideas have been taught, suggesting the types of patterns to look for in the unsolved problems Zehra gave initially.

I answered by examining those patterns:

Okay. You have not given the exact wording of the problem, because “complete the series” would mean writing an infinite number of terms (and these are sequences, not series)!

It looks like you were told perhaps to fill in a given set of blanks for the next few terms, not to give a formula; that’s helpful.

The first example is an arithmetic sequence with common difference 3, and is very easy.

The second is an alternation between two sequences: 10, 10, 10, … and 15, 50, 16, …, where you appear to have guessed (with insufficient information) that the second subsequence alternates between 15, 16, 17 and 50, 60, 70 (or maybe you would describe it differently: the process to obtain the next term alternates between using 10, adding increasing numbers 5, 6, 7, and multiplying by the same numbers). That tells me that an alternating rule is likely for the ones you asked about, which confirms my thoughts.

As I said, these are mere puzzles, and should not be graded assignments or otherwise considered to be a test of ability! In fact, better students might do worse at these, as they (like me) will be more concerned about having sufficient evidence and not just guessing.

Have you considered my suggestion for the first sequence?

I now have an idea for the second: Rather than look at the differences between successive terms, look at their ratios.

The second example is much like one I discussed last time; the apparent pattern is complicated enough that even with six terms given, a guess has to be made. To clarify what the pattern is, consider this colored version:

10, 15, 10, 50, 10, 16, 10, 60, 10, 17, 10, 70

Odd-numbered terms are all 10; even terms alternate between the sequence 15, 16, 17 (increasing by 1) and 50, 60, 70 (increasing by 10). But if only the first six terms were given, we have no actual evidence of increasing by 10! The only number in the blue sequence that was given is 50.

One possible reason for making that assumption, as I suggested, is that 15 and 16 are respectively 5 and 6 more than the 10 they follow, while 50 is 5 times the 10 it follows. That is, we have more reason to claim a pattern when we treat it not as four interleaved sequences, but as an alternation of four actions, adding a number, reverting to 10, multiplying by a number, and again reverting. But this is still entirely conjecture. So the author expects students to make such guesses, but evidently doesn’t state (in the immediate context) that other valid answers are possible.

As I pointed out, I am troubled when I see a problem that a poor student will have no trouble guessing at (being used to guessing in math), but a careful student who wants to be sure she is right will take a long time with, and maybe give up.

Zehra didn’t exactly try my suggestions; instead, she told me something I didn’t yet know:

Well, I know the answers of both as it is written in back of book

1, 2, 5, 1, 3, 6, 2, 5, 8, 4 ………. 8, 11, 7, 12, 15, 11

10, 2, 4, 40, 8, 10, ……… 100, 20, 22, 220

but I do not understand how we get these

Evidently the typo I had wondered about wasn’t, since she didn’t change anything (I think my guess had been that the 10 should be 16, so the second set of three are 4 times the first set); but now we have more information to judge the problems by. I answered, first confirming my guess for the first problem:

Okay, I can stop trying to guide you to find an answer, and just analyze the one you’ve been given.

Here are the differences I said to find for the first sequence:

1, 2, 5, 1, 3, 6, 2, 5, 8, 4
1  3 -4  2  3 -4  3  3 -4

As indicated by the colors, we alternately add 1, 2, 3, …; add 3; and subtract 4. Continuing, we add 4 to 4 to get 8, then add 3 to get 11, then subtract 4 to get 7, and so on:

1, 2, 5, 1, 3, 6, 2, 5, 8, 4,    8, 11, 7, 12, 15, 11
1  3 -4  2  3 -4  3  3 -4    4   3  -4  5   3  -4

There may be a nicer way to describe it, but my answer does agree with theirs.

Here there are enough terms given that, given the expectation of alternating additions, some constant and some changing, we could be reasonably confident that this was what was expected (but not that this is “correct”, or the only possible answer!).

Continuing:

Here are the ratios I said to find:

10,  2,  4,  40,  8,  10
1/5  2   10  1/5 5/4

We see a repeated division by 5; but we can’t be sure of anything else. But having already tried differences, we have this:

10,  2,  4,  40,  8,  10
-8  +2 +36  -32  +2

So it looks like we alternately divide by 5, then add 2, then … what? Maybe multiply by 10 — but with only one example, we can’t be sure it isn’t add 36, or something else entirely. But if we guess the multiplication by 10 (just because it feels nicer), we get this:

10,  2,  4,  40,  8,  10,    100,  20,  22,  220
÷5  +2 ×10   ÷5  +2   ×10     ÷5   +2  ×10

That gives their answer, so we are “correct”.

Here that multiplication by 10 step had no evidential support, because there were not enough terms given to support three alternating rules. Clearly, the more complicated the rule, the more data are needed to guess it confidently.

I concluded:

But I very much dislike this sort of problem: it is entirely wrong to give a “correct” answer without at least saying “other answers are possible”, and without giving an explanation for the pattern they followed, since there really is not enough evidence to make this conclusion. They are not teaching mathematics, but either (a) how to jump to a conclusion, or (b) how to feel stupid because you didn’t see the answer they wanted. But this sort of thing seems to be common (and as a mere game, it can be fun).

The important thing, again, is not to be anxious when you can’t “solve” it. It took me a while, and I have a lot of experience with these.

A pattern in the pattern?

That was the end of this thread, but the next day Zehra wrote another question, closely related to the first:

Using any proper pattern (e.g.,  ×2,  +2 , ÷2, -1, ×3, +3, ÷3, -1, ×4), write a series in following questions:

×2, +2, ÷2, ×2, ……………

×10, ÷2, ×9, ÷2, ……………

Apparently what the textbook means by “proper pattern” is one like those we have been examining, with a notation much like what I used in my last response: a sequence of operations that either repeat or change in a simple way. These questions ask the student to recognize the pattern they intend in these operations, so they are sort of a second-level pattern question. But we have the same difficulties as before. I answered:

I’m not entirely sure what they are asking for; if I were helping you in person, I would be looking at the book for an example of this kind of question, or checking the back for their answers to similar exercises, to make sure what kind of answer they want.

They may be just asking you to turn these into repetitive patterns of operations (much like those I described for the other problems):

×2, +2, ÷2×2, +2, ÷2×2, +2, ÷2, …

But their example suggests that (like the previous problems) some or all might be meant to change each time, like this (changing the addition):

×2, +2, ÷2×2, +3, ÷2×2, +4, ÷2, …

And since they said explicitly “Using any proper pattern”, it appears that anything like this would be a valid answer, and they would not give a single “correct” answer in the back.

Or maybe that’s not what they mean:

But since they said, “write a series“, not “write a pattern“, it’s also possible that they are asking for an actual sequence using this pattern; but then they would probably have given a first term so you’d have something definite to go by. For example, if the first term were 1, my first answer above would generate this sequence:

1,  2,  4,  2,  4,  6,  3,  6,  8,  4
×2, +2, ÷2, ×2, +2, ÷2, ×2, +2, ÷2, ...

while following my second pattern would yield this:

1,  2,  4,  2,  4,  7,  3.5,  7,  11,  5.5
×2, +2, ÷2, ×2, +3, ÷2,   ×2, +4,  ÷2, ...

You can try the second example.

Zehra wrote back:

×10, ÷2, ×9, ÷2, ……..

I suggested a couple possibilities:

I might just repeat all four terms over and over, with optional modification each time. That is,

×10, ÷2, ×9, ÷2×10, ÷2, ×9, ÷2×10, ÷2, ×9, ÷2, …

Since the ÷2 appears twice, we probably wouldn’t want to modify that each time (though it wouldn’t be wrong to do so, ÷2, …, ÷2, … ÷3, …, ÷3, … ÷4, …, ÷4, …). But you might choose to see the multiplier decreasing each time, as

×10, ÷2, ×9, ÷2, ×8, ÷2, ×7, ÷2, ×6, ÷2, ×5, ÷2, …

Again, I’m not very happy with the way this problem is stated, because I can’t be sure what is actually expected, and it is teaching guesswork rather than mathematical thinking; but that is what seems to be intended.

The bottom line seems to be that, if you are being taught material like this, you need to accept the fact that guessing is allowed. This can be described as inductive thinking rather than deductive thinking, and though the former doesn’t guarantee a right answer, and is not properly a part of mathematics, it is a part of science (and life); and within math, it can be used to make hypotheses (that is, fancy guesses) that you can then try to prove.

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