Induction Question (1 Viewer)

[Valentino25]

Prove by mathematical induction that n²-n is divisible by 30.
How would you go about this?
Prove n=2
Then assume n=k
Then n=k+1
Then expand and manipulate? Because that way it becomes rather large... or is there a trick i'm missing?
It's probably something obvious staring me right in the face haha.

 

[deswa1]

Can you check the question? This doesn't work for n=1,2,3,4,5,7,8...

 

[nightweaver066]

Do you mean divisible by 3 and prove for ?

 

[seanieg89]

That isn't true either.

 

[nightweaver066]

That isn't true either.

lol right. Checked wrongly. Will just wait for OP

 

[Valentino25]

Damn it. I meant n⁵-n divisible by 30.
D'oh. I have no idea why i put squared :|

 

[Timske]

Damn it. I meant n⁵-n divisible by 30.
D'oh. I have no idea why i put squared :|

Prove true for n=2

 

[barbernator]

ok ill latex up solution

 

[Trebla]

What I've got is that the expression for n = k + 1 reduces to
30M + 5k(k + 1)(k² + k + 1)
and you will need to show that k(k + 1)(k² + k + 1) is divisible by 6 (perhaps by induction as well)

 

[asianese]

This is my go

 

[nightweaver066]

Are you allowed to just explain why its divisible by 2 (without induction), then prove its divisible by 15 by induction to prove its divisible by 30?



There are two consecutive numbers here so it must be divisible by 2.

 

[D94]

Are you allowed to just explain why its divisible by 2 (without induction), then prove its divisible by 15 by induction to prove its divisible by 30?



There are two consecutive numbers here so it must be divisible by 2.

Likewise question with the number '3'. Any 3 consecutive numbers must contain a number divisible by 3. So, k(k + 1)(k - 1) are 3 consecutive numbers, so really, if this is possible, then all we need to do is prove it's divisible by 5, which is more straightforward.