Question 1203220: What is the sequence for -1, 3 , -5
Found 4 solutions by josgarithmetic, ikleyn, greenestamps, math_tutor2020:
Answer by josgarithmetic(39620)   (Show Source):
Answer by ikleyn(52802)  (Show Source):
You can put this solution on YOUR website! .
The sequence is of three terms, as you see it.
"You get what you see".
What else we can say ?
- it is not arithmetic sequence;
- it is not geometric sequence;
- it is not exponential;
- it is not periodic.
So, it is not of any sequences, that are studying in school.
O-o-o ! It is "alternate" sequence: it changes the sign from term to term,
and absolute values form arithmetic sequence; but you can not state (and even can not expect)
that it will be so with the following terms: you even don't know if the following terms do exist.
Probably, that is all that we can say.
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Do you see, the question in the post is posed in an "arbitrary" form.
It does not assume a certain answer: the answer also can be in an arbitrary form.
So, you got my answer in this "arbitrary" form,
which is, from the other point of view, absolutely precise, at the same time.
Thus, my answer is 100% adequate to your question - - - and
absolutely useless, at the same time.
So, the question and the answer are of the same worth.
But I hope, nevertheless, that you have learned something useful for yourself from my sermon.
This useful is that some questions are nonsensical, and it is better do not ask them.
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After reading this, you may ask me: how then mathematicians study sequences, predict their "next terms",
study their properties and so on ?
- - - - - - - Sequences come to Math in two ways. - - - - - - - -
One way is to determine / (to define) the sequence by a rule or by a formula.
It is the way, how arithmetic and/or geometric progressions came to Math.
For such sequences, mathematicians develop their properties based on their definitions.
Another way for a sequence to come to Math is when the sequence describes some properties
of other mathematical or physical/chemical/biological objects.
An example of such sequence is the number of diagonals of an n-gon in Geometry.
In this case the formula for the n-th term is  =  , and in this way,
we can predict any n-th term.
One more example is the Fibonacci sequence, which came to Math from consideration
of population of rabbits (i.e. from Biology).
For such sequences, mathematicians derive their properties studying the properties of relevant objects in Math or in Science.
But when somebody writes three numbers and asks "what sequence is it ?" - then we only can say
- - - it is three terms sequence which you see on the paper, or on the screen, or on the desk in classroom.
Then the question is extremely stupid, and the answer is adequate.
- - - - - - - - Hope you get my idea. - - - - - - - -
It is very important to understand it to the end (to the bottom), in order for
understand, which question about sequences does make sense and which does not.
In school, nobody teaches students to this issue, so even adult people believe that it is possible
to predict next term without having solid base for it. It is the same as to believe in our days
that the Earth is flat.
The right understanding only comes with common sense to mature mind (if you're lucky).
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
"What is the sequence for ..."
That could mean many different things. You might be looking for the next term, or the next three terms; or you might be looking for either a general formula or a recursive formula for the n-th term of the sequence.
But whatever the actual question is, it can't be answered. Any next number(s) will make a valid sequence; and of course different next numbers will mean different formulas for the n-th term.
ALL problems like this, where you are given a sequence of numbers with no other information, can't be solved, unless you know the author of the problem and can ask him what the answer is. Other than that, it is just guesswork. And that's not mathematics.
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!
If the terms are odd numbers alternating in sign, then we extend the sequence to get: -1, 3, -5, 7, -9, 11, -13, 15, ...
The formula for the nth term would be  where n starts at n = 1
The 2n-1 portion is always odd when n is an integer.
The (-1)^n portion alternates the sign from positive to negative, or vice versa.
Again this assumes the pattern extends like that.
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Another possibility is this sequence: -1, 3, -5, -25, -57, -101, -157, -225, ...
The terms in red are determined from the formula 
I used interpolation to find this formula.
For instance, plug in n = 1 to determine the 1st term
Repeat for n = 2 to arrive at  and so on.
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Here's another possibility
The starting sequence is -1, 3, -5
The gap from -1 to 3 is +4
The gap from 3 to -5 is -8
Then the jump from +4 to -8 is "times -2"
If that "times -2" pattern holds up, then the next gap could be +16 because -2*(-8) = 16
So -5 + 16 = 11 could be the next term
Then the next gap could be -2*16 = -32
So 11 + (-32) = -21
Then the next gap could be -2*(-32) = 64
So -21 + 64 = 43
-1, 3, -5, 11, -21, 43, ...
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As the other tutors have mentioned, questions that ask about the next term in a sequence tend to be too vague.
If the teacher mentioned something like "the sequence is arithmetic" or "the sequence is geometric", then it would narrow things down to be able to determine the next term(s).
In this current state, it is not enough information to simply state "-1, 3, -5" and ask for the next term(s).
Further Reading:
https://www.algebra.com/algebra/homework/Sequences-and-series/Sequences-and-series.faq.question.1195799.html
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