1. Sums of Consecutive Natural Numbers
Which natural numbers can be written as the sum of two or more consecutive natural numbers? Analyze by listing some examples.

2. 1 = ? 17 = 8+9
2 = ? 18 = or 5+6+7
3 = = 9+10
4 = ? 20 =
5 = = or or
6 = =
7 = = 11+12
8 = ? 24 = 7+8+9
9 = 4+5 or = or
10= =
11= = or or
12= =
13= = 14+15
14= = or or
15= 7+8 or = 15+16
or = ?
16= ?

3. Maybe Gauss used algebra and 1 + 2 + 3 +. . .+100 = S
the commutative property of addition = S
= 2S
100 (101) = 2S
S = 100 (101) = 5,050  
1. Can you find a formula for the sum of n
using the commutative property

4. Did you come up with
n = S
n (n-1) + (n-2) = S
(n+1) + (n+1) + (n+1) (n+1) = 2S
n (n+1) 2 or
n is the # of terms and
n+1 is the sum of the 1st & last terms
Since the last term in a series will not always be equal to n, the # of terms, and the first term in a series is not always 1, let’s rewrite the formula
n (first # + last #) 2
Can you find the sum of using this formula?

5. Did you come up with 28 (171) = 4788 = 2,394
Was there a tricky part to this problem?
How did you solve it?
What if you have to find the sum of consecutive odd numbers from 1 to 101?

6. How many numbers were we adding together? How many terms?
Did you add 1 and divide by 2 to get 51 terms?
Did you know that 51
is 2,601,
the answer that you should have come up with?

7. Take a look at these sums to see why
= = 1
= 4 = 2
= 9 = 3
= 16 = 4
= 25 = 5
The sum of odd numbers are perfect squares.