Question

Use mathematical induction to prove that the statement is true for every positive integer n. Show your work. 2 is a factor of n2 - n + 2

Answer 1

Let n = 1
then f(1) = 1^1 - 1 + 2 = 2   so it is true for n = 1
for the next number  after n   ( n+1) we have  f(n+1) =
 (n+1)^2  - (n+1) + 2 

= n^2 + 2n + 1 - n - 1 + 2 

= n^2 + n + 2 

= n(n+1) + 2

Now n(n+1) must be divisible by 2  because  either n is odd and n+1 is even OR n is even and n+1 is odd   and odd & even always = an even number.

So the function is divisible by 2  for n+1   We have shown that its true for n = 1 Therefore it must be true for n = 1,2,3,4 ... 
True for all  positive integers