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## Algebra (all content)

### Course: Algebra (all content)ย >ย Unit 18

Lesson 9: Deductive and inductive reasoning# Inductive reasoning

Sal analyzes a solution of a mathematical problem to determine whether it uses inductive reasoning. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- Is there a difference between a conjecture and an assumption?(32 votes)
- 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.(29 votes)

- So is a conjecture in math kind of the same as a hypothesis in science?(11 votes)
- 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.(14 votes)

- 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?(5 votes)- 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.(5 votes)

- what is an nth term??(2 votes)
- the variable "n" is most commonly used to symbolize ungiven numbers. any number can be represented by an "n".(7 votes)

- is conjectured similar to estimate?(2 votes)
- 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.(5 votes)

- what is the difference between conjecture, conclusion, and a justification(1 vote)
- 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(3 votes)

- I've always assumed that the "..." following a sequence implied that the pattern of the sequence continues to hold. Is this not the case? The video seems to suggest that it isn't necessarily so.(2 votes)
- I believe it does not necessarily mean that the pattern of the sequence continues but that the data set continues. We do not necessarily know what it continues to, we just know that there is more data.(1 vote)

- Is there a FORMULA or an easy way for CONJECTION of more COMPLEX Situations(2 votes)
- No, you have to use logical reasoning to solve these kinds of problems(1 vote)

- 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.(1 vote)
- 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!(2 votes)

- how do you find the conjecture or how do you find the pattern of the set of numbers?(1 vote)
- 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.(2 votes)

## Video transcript

Jill looked at the following
sequence. 0, 3, 8, 15, 24, 35. And it just keeps going, I
guess, with a dot, dot, dot. She saw that the numbers
were each 1 less than a square number. 0 is 1 less than 1, which
is a square number. 3 is 1 less than 4. 8 is 1 less than 9. 15 is 1 less than 16. Yeah, they were all 1 less
than a square number. And conjectured that the
nth number would be n squared minus 1. Now conjecture, that sounds
like a very fancy word. When someone makes a conjecture,
they conjecture, that just means that they're
making a statement that seems, or they're making a proposition
that seems likely to be true. It seems like a very reasonable
thing to say. But it's not definitely true. So she conjectured that the
nth number would be n squared minus 1. The reason why this is a
conjecture as opposed to a 100% definitely true statement,
is we don't know whether this pattern
continues. She's just going off of the
pattern that she saw so far and she just generalized it. She just assumes that
it keeps on going. But we don't know whether it
necessarily keeps going. Maybe the next number, you would
expect it to be 48, but maybe it's not 48. Maybe it's something weird. Maybe it's 2. Maybe it's 500. And so the conjecture wouldn't
hold up if you were to see that, but based on the evidence
you see so far it seems completely reasonable
that this pattern would continue. And so she conjectured that
the nth number would be n squared minus 1. Completely reasonable. Now did Jill use inductive
reasoning? Yes, she used inductive
reasoning. That's what inductive
reasoning is. You see a pattern. In this case, every term in this
sequence so far was-- if it's the third term, it
was 3 squared minus 1. The fourth term is 4
squared minus 1. The fifth term is 5
squared minus 1. So she saw the pattern and she
just generalized it to say, well, I think or I've
conjectured that the nth number will be n squared
minus 1. That's what inductive reasoning
is all about. You're not always going to be
100%, or you definitely won't be 100% sure that you're right,
that the nth number will be n squared minus 1. But based on the pattern you've
seen so far, it's a completely reasonable thing
to-- I guess you could say-- to induce.