1 = 1

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

1 + 3 + 5 + 7 + 9 = 25

a. Come up with a conjecture about the sum when you add the first <i>n</i> odd numbers. For example, when you added the first 5 odd numbers (1 + 3 + 5 + 7 + 9), what did you get? What if wanted to add the first 10 odd numbers? Or 100?

b. Can you think of a geometric interpretation of this pattern? If you start with one square and add on three more, what can you make? If you now have 4 squares and add on 5 more, what can you make?

c. Is there a similar pattern for adding the first n even numbers?

2 = 2

2 + 4 = 6

2 + 4 + 6 = 12

2 + 4 + 6 + 8 = 20

a. The formula is

**n^2**.

The sum of the first 10 odd numbers is

**100**seen on our sum of the first calculator

The sum of the first 100 odd numbers is

**10,000**seen on our sum of the first calculator

b. Geometric is 1, 4, 9 which is our

**n^2**

c. The sum of the first n even numbers is denoted as

**n(n + 1)**seen here for the first 10 numbers