Question

So I have two questions on some make up work, which is about proofs by mathematical induction.
Use Mathematical induction to prove that the statement is true for all natural numbers n
1. 5+8+11+...(3n+2)=n(3n+7)/2
2. 1+3+5+...+(2n-1)=n^2

Answer 1

Ok, so what are the steps for induction.

Answer 2

First you prove it for n=1, then you prove for k=n, and then for k+1 I think?

Answer 3

well I wouldn't say prove, you just do a base case where n=1

Answer 4

then you assume it is true for every n

Answer 5

finally you have to show that it works for n+1 using what you have assumed

Answer 6

So let's try the first one. What is the base case?

Answer 7

the base case is (3n+2)= n(3n+7)/2? I'm not completely sure

Answer 8

Base case is just say n=1 and see if the right side works, so
I like to structure mine like this

Base case: Let n=1
3n+2=3(1)+2=5 \(\huge{√}\)
\(\frac{n(3n+7)}{2}=\frac{(1)(3(1)+7)}{2}=\frac{10}{2}=5\) \(\huge{√}\)

Answer 9

See how they check out? All you are doing is testing the hypothesis

Answer 10

ok so one side has to equal the other, so the two bits in the parentheses have to balance out

Answer 11

uh yea I think you are on the right page

Answer 12

So now, you write:

Inductive step:
Assume this is true for every n. Must show it is true for n+1.

Answer 13

So do you just substitute n+1 into the parentheses where the n is?

Answer 14

well sort of, you know how we just assumed that the hypothesis was correct for all n so now we have that mumbo jumbo+ (3(n+1)+2) and we want to finangle that into if we put n+1 instead of n into that mumbo jumbo hypothesis

Answer 15

sorry, I'm a lazy typer lol

Answer 16

okay so if we solve it out we are supposed to make both sides equal again?
and no it's cool I understand :)

Answer 17

yea essentially

Answer 18

you just do some fun algebra to make them equal again, then inductively, this proves it for every n>1

Answer 19

okay, thank you so much! I will try that. Hopefully it will work out fine.

Answer 20

yea, if you want post your steps here whether it be in a picture of your work or typed and I'll check it, just tag me if you have any more problems

Answer 21

Thanks! It's really nice of you to help me. Have a good night (or day)

Answer 22

haha thanks, I'll be up for a while I have a final at 9a and need to write the essay for that time still :X but don't sweat this. Inductive proofs were the first that I learned because they have a set step by step method, eventually it just clicks. You'll get the hang of it soon

Answer 23

That doesn't sound too great ,so get some rest. All the same, best of luck on your final and essay! I think I might be getting it though, it's just a simple manner of balancing each side multiple times and showing why it works.

Answer 24

Thanks! and yea, just make sure you don't set them equal or if you do, you only manipulate the one side

Answer 25

It's just like proving the trig identities

Answer 26

so far the steps my notes have say this:
1. Show the formula works for n=1
2. Assume the statement is true for natural number k
3. Prove statement is true for k+1
I realize this different from what I guessed earlier, so I though I should post this. And trig identities were not my strong suit, I'm afraid.

Answer 27

ah, and some people use k others use n. I personally like n

Answer 28

Yes, that's what I have, but honestly I think both work

Answer 29

but yea(trig idents were not my stong suit either lol) , just follow the steps. Memorize the format and the proofs will follow

Answer 30

That's what I was having issues with, but now it's a lot clearer.

Answer 31

It's a dummy variable, doen't matter, could use gamma or ro or smiley face for all it matters

Answer 32

Well, I have to go and finish it off, so best of luck on your essay and final! And perhaps I will cover my next test in smiley faces, lol.

Answer 33

Thanks lol and feel free just make sure you define it properly :P

Answer 34

I will. Goodnight! :P