Question

A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work.
Sn: 2 is a factor of n2 + 7n

Answer 1

S1: 2 is a factor of 1^2+7*1
and so on.

Clearly, since
if n is odd, n^2 and 7n are both odd, so their sum is even
if n is even. n^2 and 7n are both even, so their sum is even.

Or,

if n is odd
(2k+1)^2+7(2k+1) = 4k^2+4k+1+14k+7 = 2(2k^2+9k+4)
and similarly for even n=2k