n(n+1)(n-1) is divisible by 3

Question

mathematical inductions to prove that statement is true

Answer 1

show that (n-1)(n)(n+1) is divisible by 3 using mathematical induction

 

show that (n-1)(n)(n+1) is divisible by 3 for n=1

(1-1)(1)(1+1)

(0)(1)(2)=0 which is divisible by 3

assume (k-1)(k)(k+1) is divisible by 3 for some k

show this is true for k+1

that is, show that (k+1-1)(k+1)(k+1+1) is divisible by 3

show (k)(k+1)(k+2) is divisible by 3

multiply the three factors together

k^3+3k^2+2k

rewrite this as...

k^3+3k^2+3k-k

group the factors

(k^3-k)+(3k^2+3k)

(k-1)(k)(k+1)=k^3-k was assumed to be divisible by 3

(k^3-k)+3(k^2+k)

(k^3-k) was assumed to be divisible by 3 and 3(k^2+k) is divisible by 3 because it has 3 as a factor

if two numbers are both divisible by 3 then their sum is divisible by 3

therefore we have shown that (k)(k+1)(k+2) is divisible by 3