Question 1

Is there a difference between a conjecture and an assumption?

Accepted Answer

As I understand it, an assumption is to take something for granted, whereas a conjecture is an educated guess based on incomplete or inconclusive evidence.

Question 2

So is a conjecture in math kind of the same as a hypothesis in science?

Accepted Answer

Yeah, kind of.  The difference is that you verify a hypothesis with an experiment and you verify a conjecture with a proof, but they both involve finding the pattern in a pile of data and predicting something that you couldn't tell from that data directly.

Question 3

I look the series `0, 3, 8, 15, 24, 35...` and have a conjecture like this: _First, +3 then +5 then +7 then +9 then +11_. Obviously, the next number should be +13: `35 + 13 = 48`. Is my induction wrong? Or should I say there are more than a reasoning which you can apply to an arithmetic series?

Accepted Answer

That seems like a reasonable conjecture based upon the pattern you've seen.  Nice inductive reasoning! :)  And you'll notice it agrees with the formula n^2 - 1 that Jill conjectured in Sal's video.

I'm not sure what you meant by your last question, though: _"Or should I say there are more than a reasoning which you can apply to an arithmetic series?"_  I will note that this sequence, incidentally, is not arithmetic, since the difference between consecutive terms is not constant.  Hope that helps some.

Question 4

what is an nth term??

Accepted Answer

the variable "n" is most commonly used to symbolize ungiven numbers. any number can be represented by an "n".

Question 5

is conjectured similar to estimate?

Accepted Answer

Sort of, but not precisely.  "Sort of" because I think you are using the word _estimate_ to mean a guess about the next value or the behavior of a sequence.  Unfortunately the word estimate also means to approximate, for example by rounding a high precision answer.

A conjecture is a statement that is `likely to be true based on what you have observed` (so far), and if what you have observed so far is accurate and really, truly does represent how the phenomenon is behaving, then your statement is exactly true, not an estimate.  If you are making observations and think you see a pattern, you can make a *conjecture* about how the next events will behave. 
An example
On Monday it rained
On Tuesday it rained
On Wednesday it rained
On Thursday it rained
So, based on those events, you can make a conjecture that it will also rain on Friday.
You cannot be too surprised, though, if the storm is over and it is sunny on Friday.

That is why Sal talked about the values represented by "..."  Those pieces of data may not follow the same pattern.  So you can *conjecture* about how they will be, but you cannot definitely prove that.  If they are different from what your conjecture predicted, then you need to re-examine what you know about ALL the known data and try for a new conjecture.

Question 6

what is the difference between  conjecture, conclusion, and a justification

Accepted Answer

Hello Dea,

A conjecture is when a person makes a statement or proposition that seems likely to be true.
A conclusion is like a judgment or decision reached by reasoning.
A justification is like the action made by someone of showing something to be right or reasonable.
Like if someone says pencils are useless and you like pencils, you would be like no pencils are good because.... (the .... is the justification)

Hope that helps!
-JK

Question 7

Is it just me or is the expression used to describe the sequence wrong? I tried calculating and it doesn't seem to work after 8.

Accepted Answer

Hi Tulsi,

The sequence matches the first six terms listed and is only supposed to fit the first six terms as we do not know what the eighth or 100th term equal to.

Hope this was helpful!

Question 8

how do you find the conjecture or how do you find the pattern of the set of numbers?

Accepted Answer

Making a conjecture is like solving a puzzle from a few of the pieces.  You examine what you know and try to determine what the pattern is.  The best way to get good at it is to practice working with sets of numbers to try to find the patterns and make the patterns into conjectures that describe the pattern.

The easiest patterns are based on differences or multiples.  
Hunting differences:
Subtract the first value from the second, the second from the third, the third from the fourth.  These are *first differences*.  Then if all the differences are the same, you already have a pattern.  If they are different, subtract the *differences* to find if there is a constant second difference. If there is, you have another pattern.
Hunting multiples:
Look for a common multiple.  Are all the terms even? Divide the second term by the first term, then the third term by the second term, and if that quotient is the same, it is a geometric sequence, possibly, if the pattern continues.  If that isn't getting you anywhere, look for alternating negatives and positives.  That could indicate a multiplication by a negative factor.
Hunting patterns:
Look to see if the term is a multiple of its n-value (item number in the sequence).  You can also check to see if one pattern applies to even terms and another pattern applies to odd terms.
Once you think you have the pattern, you need to make the conjecture which is a proposition (statement) that the number can be predicted by an equation or set of equations based on math and on the number of the term.  There are some conventions that you should follow: _n_ refers to item number (number of a term) in your statement, not to the value of the term.

Algebra (all content)

Course: Algebra (all content) > Unit 18

Inductive reasoning

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Video transcript