數學歸納法(1)

Let P(n) be the statement
"(3n+1)7^n-1=9M for all positive integer n,where M is an integer"
when n=1
(3n+1)7^n-1=27=9*3
So P(1) is true
Assume that P(k) is true
(3k+1)7^k-1=9M
when n=k+1
(3k+4)7^(k+1)-1
=7 [(3k+4)7^(k)]-1
=7 [(3k+1)7^(k)-1+3*7^(k)+1]-1
=7*9M+3*7^(k+1)+6
=7*9M+3[7^(k+1)+2]
=7*9M+3*3K [where 7^(k+1)+2=3M, it can be proved using MI and is an easy task]
=9(7M+k)
So P(k+1) is true
By the mathematical induction, for all positive integer n (3n+1)7^n-1=9M ,where M is an integer