Question

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

Answer 1

2 is a factor of n2 - n + 2

Answer 2

basis case n=1
2 is a factor of 1^2 - 1 + 2 ? yes because 1^2 -1 + 2 = 2
if 2 is a factor of k^2 -k + 1 is it true that 2 is a factor of
(k+1)^2 -(k+1) + 2 ? thats what we have to prove

Answer 3

ok?

Answer 4

okay sorry my computer shut down due to updates! @perl

Answer 5

hi

Answer 6

okay so how do we finish this?

Answer 7

so we need to prove the following statement:
"if 2 is a factor of k^2 -k + 1 for some integer k *then* is it true that 2 is a factor of (k+1)^2 -(k+1) + 2 "

Answer 8

lets look at (k+1)^2 - (k+1) + 2 , expand that

Answer 9

proofs are hard :(

Answer 10

now let me ask quick why did you expand it?

Answer 11

@perl ?

Answer 12

I think he's away...

Answer 13

we want to use the fact that
k^2 - k + 2 is divisible by 2

Answer 14

oh nevermind...

Answer 15

at some point

Answer 16

OK

Answer 17

okay than what

Answer 18

(k+1)^2 - (k+1) + 2
= k^2 + 2k + 1 - (k+1) + 2
= k^2 + 2k + 1 - k - 1 + 2
= k^2 +2k -k + 1 -1 + 2
=k^2 + 2k - k + 2
= k^2 -k + 2 + 2k
= (k^2 - k + 2) + 2k, but we know that (k^2 -k + 2) is divisible by 2 since we assumed that

Answer 19

so we have two numbers both of which are divisible by 2. therefore the sum will be divisible by 2. and we are done

Answer 20

you kinda lost me now.

Answer 21

we assumed that k^2 + -k + 2 is divisible by 2. using this assumption we can prove that (k+1)^2 - (k+1) + 2 is divisible by 2

Answer 22

okay so basically (k+1)^2 -(k+1) +2 would be my answer i dont have to put a # in for k

Answer 23

you know the statement is true for n=1. and we prove that if the statement is true for n=k, then the statement is true for n=k+1.

so the statment is true for n=1, then since it is true for n=1 it is true for n= 1+1 = 2.
now since it is true for n=2 it is true for n= 2 + 1 = 3, etc

Answer 24

well am i suppose to just write that one equation though

Answer 25

Im confused on what to put/

Answer 26

????

Answer 27

he's working on more than one problem on os apparently

Answer 28

@UsukiDoll do you know exactally what i have to put

Answer 29

>_________<!

Answer 30

I haven't taken a proof writing class, so no sorry...but I would put what perl mentioned

Answer 31

i know but would it be the equation i wrote..

Answer 32

perhaps... do you know spherical coordinates?