Clay Kitchings :: EMAT 6600 ::
Consecutive Odds and Evens (From Project Intermath)
http://intermath.coe.uga.edu/topics/algebra/patterns/r01.htm
The ÒexplanationsÓ
in this problem come strictly from algebraic proofs.
Take
the product of the middle two numbers and subtract the product of the first
number and the last number.
Example: 5, 7, 9, 11
Middle Product: 63
Outer Product: 55
63 – 55 = 8
Example: -9, -7, -5, -3
Middle Product: 35
Outer Product: 27
35 – 27 = 8
Conjecture: This difference will always be 8.
Proof:
Consider
four consecutive odd numbers:
(2k+1),
(2k+3), (2k+5), (2k+7), where k is an integer.
Middle
Product: 
Outer Product: 
Subtract: 
Does the same thing happen for even numbers?
Indeed
it does. Consider the following four consecutive even integers:
2k, 2k+2, 2k+4, 2k+6, where k is an integer.
Middle
Product: 
Outer
Product: 
Subtract:
What happens for consecutive integers in
general?
Consider
the following consecutive integers:
k, k+1, k+2, k+3, where k is an integer
Middle
Product: 
Outer
Product: 
When
the two are subtracted, 2 is always the result.