Main content

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

Lesson 9: Deductive and inductive reasoning# Using inductive reasoning

Sal uses inductive reasoning to find an expression for the nth number in the sequence 6, 9, 12, 15,.... Created by Sal Khan.

## Want to join the conversation?

- I thought about (3 (n - 1)) + 6.(6 votes)
- That is because when you expand 3(n - 1) + 6, you get 3n - 3 + 6 which is equal to 3n + 3. That is the same as 3(n + 1) = 3n + 3.(13 votes)

- what does "nth" number mean?(4 votes)
- It's like a stand-in for any number... so a variable. If that shortens things. :-)(2 votes)

- Isn't this reasoning deductive? Couldn't we say that Sal "proved" that the nth term is 3n+3? To put it in another way: how can we say that Sal conjectured that the Nth term is 3n+3?(2 votes)
- It is inductive because it is based upon observing the pattern in the given numbers. Conclusions based on observations are inductive. Sal to specific observations and used them to draw a general conclusions.

Deductive reasoning is when you start with a general rule(s) and you draw a specific conclusion.

Hope this helps.(5 votes)

- Why are simple math concepts such as skip counting involved in precalculus?(2 votes)
- And arithmetic progressions(AP) where the difference between two consecutive terms are constant, such as:10,15,20,25....

Then to find out the nth term of an AP where the first term is 'a' and the constant difference is 'd' then the "n"th term is : a+(n-1)*d.

Finding the sum is easy also like: when "n" is the number of terms, 'I' is the first term and "F" is the final term, the sum is :

[n(I+F)]/2(2 votes)

- Why is 3n+3 the right anwser while the pattern is adding by 3 so wouldn't n+3 work too?(3 votes)
- Try using the explicit or recursive definitions for arithmetic sequences https://www.khanacademy.org/math/trigonometry/seq_induction/seq_and_series/v/arithmetic-sequences.(2 votes)

- what is the next number, 64,-16,4,-1,
*__*(3 votes)- 1/4, as each subsequent number in the series is a square of half the previous number's square root, alternatively positive, then negative.(2 votes)

- What does nth mean is that a number?(1 vote)
- When we are working with sequences (and series), we care about the value of each term AND the number of the term. In the following sequence, the first term has a value of 4 and n=1

The second term has a value of 6 and n=2 (the number of the term is 2)

The nth term means the term with a number of n. If n=2, we are asking for the second term. If n = 6, we are asking for the nth term, which I did not give below. You can write a formula to allow you to tell me any value if I give you any value of n.

Here is the sequence: 4 6 8 10 12 ...`Term 1 = 4`

n = 1`Term 2 = 6`

n = 2`Term 3 = 8`

n = 3

etc.

You can build a formula to tell what other values of terms would be.

a_n = amount you want to find out`nth term`

a₁ = first amount or amount of the first term which was 4

d = common difference which was 6 - 4 = 2

a_n = a₁ + d(n-1) → formula for`nth term`

if it is arithmetic

a_n = 4 + 2(n-1) → formula for`nth term`

Or we can say that a_n = 2n + 2

(the same thing as the other formula, but simplified)

So, if I want the 50th term (n = 50)

a_50 = 4 + 2(50 - 1) = 4 + 98 = 102

With the other formula, a_n = 2n + 2 = 2∙50 + 2 = 102

a_50 = 102

As you study sequences and series, you will see that there are many kinds of sequences--this is just a very simple example.(1 vote)

- There used to be some really nice assignments/ practice that went w. deductive reasoning ("logical arguments and deductive reasoning" was the name)a couple years ago. Is this content still available elsewhere? It helped my students to iron out their issues.(1 vote)
- When they restructured the menus, they left some content stranded (not on any menu). I have found items using the search bar even when they aren't on a menu. Give it a try and see if you can find the assignments/videos that you want.(1 vote)

- I still don't understand why you have to add one from the term to the number.(1 vote)
- What will be the next number for these sequence 9,4,3,12,37,84,_ ?(1 vote)

## Video transcript

What is the nth number in
the following sequence? Let's see. The first term here is a 6. So over here I'll
put the term. So this is the number. I'll just write num there. That's the number. And this is the term. So the first term here is a 6. It looks like we add 3 to
that to get to a 9. Or maybe there's some
other pattern here. So that's the second
term, is 9. Then they go from 9 to 12. Looks like we add 3 again. So we keep adding 3. So the third term is 12,
the fourth term is 15. And we need to think of a
general expression that will give us the value
of the nth term. The nth number in the
following sequence. So let's think a little
bit about it. We know that we're incrementing
by 3 every time and we're starting at 6. So let's write these in terms
of multiples of 3, since we know that that's the
increment level. This first term right
here is 3 times 2. The second term right
here is 3 times 3. This third term is 3 times 4. And so it looks like it's 3
times one more than the actual term you are dealing with. Right? Here, if you take your term,
you add 1 and then multiply that by 3, you get 6. Take your term, add 1, then
multiply it by 3. Take your term, add
1, you get 4. Then multiply by 3. Take your term, add 1-- which
would be 5-- and then multiply by 3, you get 15. So it looks like any time-- so
if you have your nth term. So if you go all the way out
here to your nth term, the value here would be
you'd add 1 to it. So you'd have n plus 1, and then
you'd multiply that by 3. So the nth term is
3 times n plus 1. Or if you wanted to distribute
out the 3, it'd be 3n plus 3. And just make sure that
works for all of them. The first term, 3 times 1
is 3 plus 3, that works. That's 6. The second term, 3 times
2 plus 3 is 9. So it works for all
of these terms. So that is the nth term. We have to assume that this
pattern continues.