Question

Use mathematical induction to prove that the statement is true for every positive integer n.

2 is a factor of n2 - n + 2

I really need help on this one

Answer 1

I don't understand what the question is asking me to do. Can anyone shed some light on it.

Answer 2

Have you learnt anything to do with "mathematical induction" before?

Answer 3

Yes but im not good at it. Could you help me.

Answer 4

What have you done so far with the question?

Answer 5

Nothing. to be honest. Mathematical induction is hard for me. I keep getting it wrong. so i came here for help. I dont even think im starting it right.

Answer 6

What would your first step be?

Answer 7

So you know there are steps to mathematical induction right? Step 1, Step 2, etc.

Answer 8

So what would Step 1 be?

Answer 9

Step 1 is always first proving that what you're given is correct. So you prove that n=1 first.

Answer 10

Oh. You dont start off with 2?

Answer 11

If the question said prove that this.....for n>1 then you start off by proving n=2. But in this case, you prove n=1.

Answer 12

okay

Answer 13

I just wanna ask can't you say that n^2-n+2 is divisible by 3. Then prove it that way?

Answer 14

Part of your question states: "the statement is true for every positive integer n." That should be enough for you to know that you start off by proving n=1.

Answer 15

so 1 - 1+2 = 2?

Answer 16

Okay, so you write For n=1. ANd yes. Well done.

Answer 17

And then you write which is a factor of 2.

Answer 18

THen you can write
"therefore true for n=1"

Answer 19

That will be your first step.

Answer 20

i meant divisible by 2

Answer 21

4-2+2=4

Answer 22

Your 2nd step would be: "Assume that the formula is true for n=k" or somethign along the lines of what I wrote.

Answer 23

You don't need to go and write for n=2. You already proved for 1.

Answer 24

That would be a waste of time. You only prove for n=2 if n=1 was a trivial proof.

Answer 25

Trivial meaning, very obvious that it works for n=1. So you would have to prove for n=2. But in this case, it isn't trivial.

Answer 26

oh okay

Answer 27

Thank you!

Answer 28

Can I ask you one more?

Answer 29

No worries, so your second step would be writing the whole expression using k.

Answer 30

Do you know the rest of your steps for mathemetical induction?

Answer 31

Or did you think that the step 1 was enough to prove your question?

Answer 32

Because there are numerous steps in mathematical induction, not just one step and you're done.

Answer 33

I believe step 1 was enough

Answer 34

unless you think it would be wise to do the other steps

Answer 35

Have you learnt how to do questions involving mathematical induction, in class before? Or have you not paid full attention or something? Because mathematical inductions requires several steps to do in order to complete your question.

Answer 36

induction*

Answer 37

I have to teach myself.

Answer 38

Okay, well let me tell you that Mathematical inductions requires about 3-5 steps before you finish proving your question.

Answer 39

Can I send you the other ones thru a message and you can help me with though. Im coming up with some ridiculous answers. and yes plz

Answer 40

Uh, I think you should do the other ones yourself, but I can help you with this one.

Answer 41

Lol. if thats what you feel is right. ive tried since this morning. lol

Answer 42

Most of the times, these questions use the same steps. The only step that changes is step 3 which is where you do your algebra.

Answer 43

ANd I feel you don't fully grasp what mathematical induction is.

Answer 44

So let's continue on with Step 2.

Answer 45

Like I said before, write the expression given to you in your question: "n^2-n+2" in terms of k. SO substitute n for k.

Answer 46

ok

Answer 47

Could you do that for me and show me the expression in terms of k please?

Answer 48

I dont get it. It only asked for 2. where does the other things go?

Answer 49

What do you mean 2?

Answer 50

To my knowledge, these questions require about 1 page of working.

Answer 51

Look at my question "Use mathematical induction to prove that the statement is true for every positive integer n. 2 is a factor of n2 - n + 2"

Answer 52

To prove that this statement is true for ALL positive n integers, you need to cover these steps I'm willing to give right now.

Answer 53

wouldnt K be 1,2,3 ect. but it doesnt ask for that.

Answer 54

I looked at your question. When a question uses the words mathematical induction, you would instantly think this questions requires you to do 1 page of working because it's mathematical induction.

Answer 55

these types of questions*

Answer 56

but it asked for the 2 nothing else. Im not saying your wrong. Im just really confused.

Answer 57

You're confused because you're trying to learn something that is completely unknown to you. I'm telling you everything about "Mathematical Induction" which is what you're required to know.

Answer 58

It asked whether that statement is true. In order to prove that statement using Mathematical induction, there are steps required for you to prove that.

Answer 59

what is the K than?

Answer 60

Now are you ready to show me the expression in terms of k.

Answer 61

k is another variable you use to show that if it works for k and k+1, then it will work for k+2 and so on. After that if both k and n are in the same form, then you proved that the statement is true for all positive integers "n"

Answer 62

so we was suppose to do k before n?

Answer 63

nvm read that wrong. its late here

Answer 64

so how would i lay out the k formula

Answer 65

Do I need to repeat myself over and over? I wrote it above befoer you said you were confused and everything.

Answer 66

before*

Answer 67

I see that. but what wouldnt it be the same as it would with n? I dont get that

Answer 68

why not what*

Answer 69

That's how mathematical induction works. If you want to complain to the people who invented this, then go ahead. I'm just showing you the way to do mathematical induction.

Answer 70

but what is the difference

Answer 71

between k and n?

Answer 72

Because if you say if n=n, then you're going to get confused even more.

Answer 73

k^2-k+2
right?

Answer 74

Yes. Now we write:
For n=k+1.

Answer 75

but you should write =2m.

Answer 76

at the end of your expression.

Answer 77

And say that m is an element of Real numbers or Integers.

Answer 78

\[k^2-k+2=2m\]

Answer 79

okay

Answer 80

And then you write For n=k+1.
Beneath that, you start off by saying LHS= (k+1)^2-(k+1)+2

Answer 81

Because you're now substituting k for k+1, to see whether it works for the next term.

Answer 82

Now, we need to "RIG" that LHS into the same form as "k^2-k+2"

Answer 83

\[\large LHS=(k+1)^2-(k+1)+2\]
Okay, so in order to turn this into the same form as the original expression, we need to expand/distribute the (k+1)^2 and get a k^2 from that.

Answer 84

\[\large LHS=k^2+2k+1-(k+1)+2\]
Could you simplify that for me please?

Answer 85

One sec.

Answer 86

LHS =\[LHS=4+4+1-(2+1)+2\]

Answer 87

You don't substitute k with any numbers....I only asked you to simplify.

Answer 88

DO you know what simplifying is?

Answer 89

Thanks for your help Azteck. I think I need some rest. and a break.

Answer 90

For a better learning experience, it's best to ask a question when it's not midnight.

Answer 91

And no worries. Please try and ask your questions when you're actually alive and ready to learn insteadof being tired and ready for bed. ;)

Answer 92

Ive been on this everysince i woke up. And I will try that lol

Answer 93

still willing to help azteck?