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This question is taken from 11th class Math book. Look at this question:

im

At the very first glance one can tell that all the three sequences are G.P But! by using interpolation(as this answer explains) we can give any number of answers to (a),(b) and (c).

  • Is the above mentioned question technically, a mathematical question ?

The above mentioned question is taken from NCERT, Mathematics textbook for Class XI, chapter-9 Exercise 9.3, 5th question.

Can we use interpolation to solve this question on sequences? The book mentions only one correct answer for 5th question. If The question-5 that I quoted in the question above can also be answered by interpolation then this way the book will be technically incorrect; which is less likely to be because it is a standard book.

So, is the book technically incorrect?

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    $\begingroup$ possible duplicate of sequence of numbers $\endgroup$ – user21820 Jun 6 '14 at 6:31
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    $\begingroup$ Just because it is a textbook does not mean everything is correct, and even if it is correct, it does not mean that there is no other correct answer. For this kind of problem, as you know there is no correct answer (even if we use Kolmogorov complexity) unless we specify the encoding, so we can't even talk about "technical correctness". If the book insists that there is no other correct answer, then we can safely say that the book is absolutely wrong. And this kind of question is just about as mathematical as "What is the most common integer?". $\endgroup$ – user21820 Jun 6 '14 at 6:37
  • $\begingroup$ @user21820 fun fact the most common random integer with two digits specified by a general person is $37$ (at least a few years ago it was). $\endgroup$ – DanZimm Jun 6 '14 at 6:43
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    $\begingroup$ This has been discussed on meta already. It is good manners to link to old versions of questions! $\endgroup$ – user1729 Jun 6 '14 at 8:33
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    $\begingroup$ Also also, I do not agree that this is a duplicate of the cited question @user21820. The same topics are discussed, yes, but this is a more philosophical question. The OP clearly knows about the paradox of your cited question and is asking "therefore, is this maths?". $\endgroup$ – user1729 Jun 6 '14 at 8:39
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I've already answered the actual question in my comments, but I'd like to answer the question in the textbook. I think (b) and (c) are not geometric progressions. In each case, just multiply the previous two to get the next one.

In case you didn't realize, I'm joking. =)

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  • $\begingroup$ Thankyou for your answer and constructive comments. I personally agree with you that the question is ill-posed. The book is a standard book in India. IT is used to teach every 11th class student in India. Should I email to the book authors telling them that there's a technical mistake in their book? $\endgroup$ – user103816 Jun 6 '14 at 7:42
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    $\begingroup$ @user31782: There isn't a technical mistake, and sometimes these questions are just for fun, but if I am the author I definitely will put a note to that effect, at the very least saying that there is no correct answer. Sometimes fun is good, and finding patterns is fun, but we can distinguish that from correctness. Games are fun too, but neither correct nor wrong! $\endgroup$ – user21820 Jun 6 '14 at 7:44
  • $\begingroup$ The book doesn't put any note. In the answer it says: 5.a) 13th, b) 12th, c) 9th. All the questions except this one mentions that the sequence is G.P. I think the question is ambiguous. It's nothing about fun. Across whole India this book is used as a reference. It is not appropriate to teach wrong things. Students like me are dangerously misguided due to these kind of ambiguous books. $\endgroup$ – user103816 Jun 6 '14 at 7:53
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    $\begingroup$ @user31782: Then perhaps you can just send a note to the authors that maybe the question should say "Which term of the following geometric progressions is: (a) 128 in ...". $\endgroup$ – user21820 Jun 6 '14 at 9:41
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As this question is trying to be pedantic, then yes, the answers are not quite right in the sense that the given solution is not unique. However, such question do have a place in a maths text book. To quote Jim Belk's find answer to a very related question on MathEducators.SE,

Theoretically, there's no way to determine the next term in the sequence $$ 1,\quad 2,\quad 4,\quad 8,\quad 16,\quad\ldots $$ It literally could be anything.

At the same time, it is a vitally important skill to be able to look at this sequence and say "it looks like the powers of 2". This answer is correct in the sense that any mathematician looking at this sequence would have that response, and a student who doesn't have that response when looking at this sequence has a serious gap in their knowledge.

These questions are not about giving the correct answer, but about "sensing" what the answer should be. This is an important skill. I suggest that you read his whole answer, which can be found here.

In the comments to this answer, and elsewhere in this thread, the idea of placing a disclaimer in the text's answers (pointing out that any solution can be justified) is suggested. However, most students who are reading the book will not be very mathematically mature. This disclaimer will confuse them, and it will shift the point of the question: The question is about pattern recognition, and these confused students will be trying to work out what on earth this disclaimer is going on about! Which is not a good thing. A better state of affairs would be for a teacher to point out this discrepancy to the better students in a class, and then these students can have some fun trying to work out different patterns, and hence increase their understanding. This means that the good students are encouraged more, while the poorer students are not sidetracked into confusion.

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  • $\begingroup$ Thankyou for your answer. I've already read those posts yesterday. I understand that these type of question are ambiguous. The only point of my question is to check whether that book is technically correct or not. Usually I believe in books very much, especially Math books. The book doesn't mention that the question is about pattern recognition. Before coming to MSE I had a firm believe that these type of questions have unique answer. $\endgroup$ – user103816 Jun 6 '14 at 9:00
  • $\begingroup$ Then your mathematical maturity is growing! However, most students who are reading the book will be less mathematically mature than yourself. It will just be confusing to them to point out that any solution can be justified, which is a bad state of affairs. I will add this comment to my post. $\endgroup$ – user1729 Jun 6 '14 at 9:02
  • $\begingroup$ I agree that some students may find it confusing to know the truth. I'd argue that the author of these kind of books should put this disclaimer: "This book is for immature students". There are some people who only study through books(self-study!). I'm a kinda student. I'd wary from reading these kind of books. I'm not criticizing your point. I am giving my opinion. $\endgroup$ – user103816 Jun 6 '14 at 9:17
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    $\begingroup$ Related. I draw parallel's between Colonel Nathan R. Jessup and the author of your book. $\endgroup$ – user1729 Jun 6 '14 at 9:31
  • $\begingroup$ I'd say that such questions have a place in a recreational maths book, but so would "2, 12, 1112, 3112, 132112, ?". As for a textbook, such open-ended exploration of sequences can surely be part of the text, but there should be a clear distinction between a hypothesis and a theorem. For your example of "1, 2, 4, 8, 16, ...", a hypothesis can be that it may be powers of 2, but as you should know it could also be mathworld.wolfram.com/CircleDivisionbyChords.html, so it is important to make it clear that it is a hypothesis and not justified when without a proof. $\endgroup$ – user21820 Jun 6 '14 at 9:58
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It depends how you define a mathematical question. It's mathematical in the sense that there are numbers, patterns and the such. Then again there's a good joke that shows how everything comes down to math and thus down to philosophy.

Nonetheless I think the question you're asking is as follows:

Does question $5$ have a well defined unique answer?

This is still somewhat of a subjective question. If you're in a calculus class and you've been studying sequences then in some sense, yes, there is a unique well defined answer as the students probably only know about simple sequences, i.e. geometric progressions. This leads us to questioning what to do about the students (like me) who would be smart a**** and give their favorite number as an answer and an explanation as

Really we can have this number appear at any term besides the first few that are listed since the sequence isn't defined as a formula, thus we can assume that any term of the sequence is any value we want.

This is where the subjectivity comes out; that is, I personally would say this person clearly understood the point of the question and thus deserves full credit. Although, others may think that they shouldn't get any credit since it's not the answer the teacher was looking for for a calculus class.

Anyhow if this problem was given in a more advanced class I think that reasoning above is valid, thus the question was either a trick question or was ill-posed.

At least these are my thoughts on the matter at hand, I'm open to disagreement.

Nonetheless if we were given, for example for $5a)$ that the sequence $a_n$ was defined as $$ a_n := \left( \sqrt{2}\right)^{n} \; \forall n \in \mathbb{N} $$ then I would say the question is well posed, since every term of the sequence is accurately defined (to some extent).

In the end I would say whether or not this is a well-posed problem comes down to context. I personally don't find it to be very constructive the way it is, and thus I don't think it's a good question, but again that's subjectivity.

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    $\begingroup$ I don't think we should be so forgiving and say that "Does question 5 have a unique answer" is subjective, since the answer is "Obviously not, otherwise you can prove that the answer is unique!". =) $\endgroup$ – user21820 Jun 6 '14 at 6:58
  • $\begingroup$ @user21820, I'm confused. Do you mean "If an answer is unique, then it is provably unique"? Is this a theorem (or axiom???) in some formal logic system? $\endgroup$ – achille hui Jun 6 '14 at 7:25
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    $\begingroup$ @achillehui: When people say "It is unique.", they usually mean "You will have to accept that it is unique.", in which case they should justify it with a proof. Otherwise I can just say, "No.". $\endgroup$ – user21820 Jun 6 '14 at 7:38
  • $\begingroup$ @user21820, Oh I see, thanks. $\endgroup$ – achille hui Jun 6 '14 at 7:41
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    $\begingroup$ @achillehui: Anyway I was partly joking, but the point is that many people don't realize the importance of justifying their answer. Finding a pattern is insufficient justification for concluding that the answer is unique. $\endgroup$ – user21820 Jun 6 '14 at 7:42

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