Take an integer n. The next consecutive integer is n + 1
Subtract the difference of the squares:
(n + 1)^2 - n^2
n^2 + 2n + 1 - n^2
n^2 terms cancel, we get:
2n + 1
2 is even. For n, if we use an even:
we have even * even = Even
Add 1 we have Odd
2 is even. For n, if we use an odd:
we have even * odd = Even
Add 1 we have Odd
Since both cases are odd, we've proven our statement.
Subtract the difference of the squares:
(n + 1)^2 - n^2
n^2 + 2n + 1 - n^2
n^2 terms cancel, we get:
2n + 1
2 is even. For n, if we use an even:
we have even * even = Even
Add 1 we have Odd
2 is even. For n, if we use an odd:
we have even * odd = Even
Add 1 we have Odd
Since both cases are odd, we've proven our statement.