Use mathematical induction to prove that 6 divides , for all natural numbers. (For example, n=0..1..2)
Im confused on how to set this up.
Hi
As you've seen, it's easy to check that divides when
Now, to prove that for any in we have to assume that for some integer divides and then prove that divides
We can write:
Hence
An integer is divided by iff it is divided by and and a prime divides a product iff it divides at least a factor.
If and divide or then they divide
If divides it also divides and then divides
If divides it also divides and then divides
Therefore
Of course, writing gives an immediate but non inductive proof.
Another proof by induction (maybe more comon, try to do it) consists in, when you assume that divides develop and you find something which is a sum of and an integer If you show that is divided by then divides , and you can end your proof.
Hello tokioThis is the third (and there's a fourth) induction question you've posted in the last couple of days. We're here to help you learn how to succeed at Mathematics, not to do your homework for you.
I'll start you off:
is the propositional function: divides
Now look at the expression , and, by removing the brackets, simpifying and re-arranging the terms, see if you can prove that ; in other words, that divides .
Show us your working, and, if you can't do it, we'll see about the next stage.
Grandad
PS I see someone's beaten me to it. But let's see some working next time.
PPS Since you have been given a solution, here's an easier one:
Now one of and is always even. So is divisible by . So is divisible by .
is is divisible by , which is true. So is true .