Mathematical Induction (1 Viewer)

[izi]

Hey guys can u pleez help me out on this one, its my assessment task question:

"Prove by mathematical induction that (n^3+2n) is dividable by 3 for all integers n" including a written solution

thanx heaps

 

[withoutaface]

Prove for n=1

1^3+2(1)=1+2=3 which is clearly divisible by 3

Assume for n=k, where k is a positive integer

k^3+2k=3p (where p is an integer)

Prove for n=k+1
(k+1)^3+2k+2=(k+1)^3+3p-k^3+2=k^3+3k^2+3k+1-k^3+2+3p=3(k^3+k^2+k+1+p) which is clearly divisible by 3.

:. divisible by 3 for all positive integers n



Prove for integers n<0

if n^3+2n=3p for all integers n>0

(-n)^3-2n=-(n^3+2n)=-3p which is divisble by 3

:. true for all integers n<0

Prove for n=0

0^3+2(0)=0 which is divisble by 3

:. true for all integers n

 

[izi]

is 0 an integer?

 

[withoutaface]

is 0 an integer?

Yep<tenchars>