Clay Kitchings :: EMAT 6600 :: Consecutive Odds and Evens (From Project Intermath)

 

 

 

http://intermath.coe.uga.edu/topics/algebra/patterns/r01.htm

 

The “explanations” 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.