Let p(n) be the statement
n^3+5n is divisble by 6
For n=1
(1)^3+5(1)
=1+5
=6 , which is divisble by 6
therefore n=1 is true
Assume p(k) is true for some positive integers
k^3+5k = 6M where M is integer
k^3=6M-5k
for n=k+1
(k+1)^3+5(k+1)
=k^3+3k^2+3k+1+5k+1
=6M-5k+3k^2+3k+1+5k+1
=6M+3k^2+3k+6
Because 3k^2+3k = 6N , where N is integer
=6M+6N+6
=6(M+N+1) which is divisble by 6
therefore p(k+1) is true
therefore by the princlpe of mathematical induction
n^3+5n is true for all positive integers
少許難....難在紅part
