Question

prove 2 is a factor of (n+1)(n+2) for all positive integers

Answer 1

To prove that 2 is a factor of (n+1) (n+2).

We shall prove prove by using mathematical induction as follows:

Step 1:

For n = 1, we get:

(1 + 1) (1 + 2) =2 X 3 = 6.

Since 6 is divisible by 2, the result is true for n = 1

Step 2:

Assume the result is true for n = k.

i.e.,

Let

2 be a factor of (k + 1) (k + 2)

So, we get:

(k +1) (k + 2) = 2 r                              

i.e.,

k² + 3k + 2 = 2r            (1)

for some integer r

Step 3:

To prove the result is true for n = k + 1

i.e.,

To prove:

2 is a factor of (k + 1 + 1) (k + 1 + 2)

(k + 2) (k + 3) = k² + 5k + 6

                  = (k² + 3k + 2) + 2k + 6

We note:

On the RHS, 2 is a factor of (k² + 3k + 2) by equation (1).

2 is a factor 2k.

2 is a factor 6

Thus, we have proved:

2 is a factor of (k + 1 + 1) (k + 1 + 2)

i.e.,

We have proved the result is true for n = k + 1 if it is true for n = k.

Thus, by mathematical induction, we get the result:

2 is a factor of (n +1) (n + 2)