Even & Odd Numbers

The natural numbers have several common subsets, such as even and odd numbers and composite and prime numbers. Mathematicians often need to use even or odd numbers in proofs. Even numbers are denoted 2n and odd numbers 2n-1 or 2n+1. These are guaranteed to be even or odd, respectively, no matter what n is equal to. Use this idea to prove the following situations:

ACTIVITY: Consider the subset of odd numbers {1, 3, 5, 7, 9, 11, 13, ...}. Add the first five pairs of consecutive odd numbers and record each sum. What is happening and why? Algebraically prove your conjecture for the nth odd number pair.

ACTIVITY: Sum the first two odd numbers. Sum the first three odd numbers. Sum the first four odd numbers. What is happening and why? Prove your conjecture for the nth sum of odd numbers. There is also a geometric proof of this activity.