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## Algebra (all content)

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

Lesson 9: Deductive and inductive reasoning

# Inductive reasoning (example 2)

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?

• So would this be deductive ? • At first it says Luis, but later in the question it says Carlos...
Did anybody else catch that? • Can anyone exploain to me how to do thes problems? I am trying to wrap my mind and still cannot get it...... :( I really am bad with inductive reasoning.....

i know the pattern but i do not know how to write a conjecture

&
1/2,1/4,1/8... ( as each number progresses.... 2^previous+1)
&
-3,6,-9,12... (- or + depending..... (previous number + or - [previous subtraction or additon -or+ 3])
(1 vote) • The question is old but I will answer it anyway. Just for fun (:

The first sequence 2, 4, 16, 256 , 65536, … can be written as:

2^1, 2^2, 2^4, 2^8, 2^16, …

If we only take care of the exponent, they go in this manner:

1, 2, 4, 8, 16, which is the same as:

2^0, 2^1, 2^2, 2^3, 2^4, …

As it can be noted, the 1st of these is raised to 1 – 1 = 0. The 2nd one to 2 – 1 = 0. Therefore, the nth terms in this sequence (not the original), will be a 2 raised to the n – 1 power:

2^(n-1)

Since 2^(n-1) is the exponent of the nth term in the original sequence, we can place it in that one, but being careful of putting it between parenthesis, otherwise the rule of exponent will take 2^(n-1) as if it were 2*(n-1):

2^[2^(n-1)]

In the second sequence: 1/2, 1/4, 1/8, ... if we just look at the denominators, the 1st is 2^1, the 2nd is 2^2 and so on. And so the nth term would be 2^n. But how do we get the 1 as numerator? By rule of exponent, we know that a^(-b) = (1/a)^b. Thus the nth term in this sequence is:

2^(-n)

In the third sequence: -3, 6, -9, 12, ... it looks like the 2nd term is the 1st one multiplied by 2, the 3rd terms is the 1st again times 3, and so on. This part can be assumed to be 3*n for the nth term of the sequence. Now, the first term looks like if it were multiplied by -1, the 2nd by 1, the 3rd by -1 and so on. We can conjecture that it’s a positive 1 when even and a negative 1 when odd. -1 raised to an even/odd power gives us this same pattern. So the nth term of the sequence is:

(-1)^n * 3n
• Isn't it inductive reasoning because Luis conjectured the answer he found for this problem to be to for ALL x and y? Not just the x and y in this problem. It seems he is making a inference about what x and y represent in general.
(1 vote) • I'm learning a lot about inductive vs deductive reasoning. I would like to practice this concept. The test I need to take involves both reasonings. Is there a place I can go to practice this concept? • Around Carlos said that I conjecture that the expression=x^3-y^3. Then Sal said that it wasn't a conjecture because a conjecture is making a generalization. However couldn't he be making a generalization for all numbers, as he didn't prove the result for imaginary numbers?
(1 vote) • Yes and no. First, Sal is correct, it's not a conjecture.
Think of it this way, a conjecture is basically you saying, based on what we have, I think that this will be true, but I'm not sure.
So, if at the begining he said, I conjecture that (stuff) = x^3 - y^3, but didn't prove it, then that is a conjecture.
Now since he went on and made a proof using algebraic manipulation, it is no longer a conjecture about something that is unknown, but it is now a proof.
A conjecture can be proved or disproved because it is unknown at the time, but a proof, if done correctly, is 100% fact and nothing you say or I say will change the fact that it is correct.

To answer why it is general... Yes, it is the general case for all x and y that the function will be of the form x^3 - y^3, but that doesn't make it a conjecture. It is a generalization based on a proof.
• For this example I understand that inductive reasoning was used, however why was the letter "n" used in the example for Inductive reasoning? Could we have used "x", or "y" or does the "n" reflect the word "number"? • Show example of inductive pattern   