Is this question of sequence a Mathematical one, i.e. does it have objectively only one answer for each subpart.

Question

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

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?

Answer 1

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. =)

Answer 2

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.

Answer 3

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.