# Sum of n numbers

(On the board) Find the following sums:

```1 + 2 =
1 + 2 + 3 =
1 + 2 + 3 + 4 =
1 + 2 + 3 + 4 + 5 =
1 + 2 + 3 + 4 + 5 + 6 =
1 + 2 + 3 + 4 + 5 + 6 +7 =
1 + 2 + 3 + 4 + 5 + 6 +7 + 8 =
1 + 2 + 3 + 4 + 5 + 6 +7 + 8 + 9 = ```

What is the sum of the first ten numbers?

What is the sum of the first eleven numbers?

What is the sum of the first twelve numbers?

Is there a shortcut to finding these sums?

What is the shortcut?

Here is a suggestion: Line a string of numbers up backwards under the original string

``` 1 +  2 +  3 +  4 +  5 +  6 +  7 +  8 +  9
9 +  8 +  7 +  6 +  5 +  4 +  3 +  2 +  1
10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 ```

Try using this idea to find the sum of the first thirteen, fourteen, and fifteen numbers.

Develop this idea with the class. Have groups work on reducing this to a formula.

Try not to reveal the formula: push the class towards higher numbers of sums. The formula is: (the highest number)(the highest number plus one) then divide by two

```n(n+1)/2 or n² + n
2```

This is called a quadratic equation or second degree equation. What is the greatest common factor? Factor it out.

Produce a table of values of for y = (n² + n)/2 from -5 to 5. Make a graph with n on the x-axis and y on the y-axis. Note that graphing has NOT been specifically taught prior to this point, unless the golf ball labs preceded this laboratory. This work proceeds with the assumption that students have done math in the past. During such a class I circulate and provide individual assistance as necessary.

Homework:

```1 + 3 =
1 + 3 + 5 =
1 + 3 + 5 + 7 =
1 + 3 + 5 + 7 +9 =
1 + 3 + 3 + 7 +9 + 11 =
1 + 3 + 3 + 7 +9 + 11 +13 = ```

Is there a formula for the sum of odd numbers?

What is the formula for the sum of odd numbers?

```2 + 4 =
2 + 4 + 6 =
2 + 4 + 6 + 8 =
2 + 4 + 6 + 8 +10 =
2 + 4 + 6 + 8 +10 + 12 =
2 + 4 + 6 + 8 +10 + 12 +14 = ```

What is the formula for the sum of even numbers?

Fill in the following table:

 Substitute the below into (n² + n)/2 Answer What is the sum of the first zero numbers? 0 _____ What is the sum of the first one number? 1 _____ What is the sum of the first two numbers? 1+2 2 _____ What is the sum of the first three numbers? 1+2+3 3 _____ What is the sum of the first ten numbers? 1+…+10 10 _____ What is the sum of the first twenty numbers? 20 _____ What is the sum of the first thirty numbers? 30 _____ What is the sum of the first hundred numbers? 100 _____ What is the sum of the first two hundred numbers? 200 _____ What is the sum of the first three hundred numbers? 300 _____ What is the sum of the first thousand numbers? 1000 _____

Developed by Dana Lee Ling with the support and funding of a U.S. Department of Education Title III grant and the support of the College of Micronesia - FSM. Notebook material ©1996 College of Micronesia - FSM. For further information on this project, contact dleeling@comfsm.fm Designed and run on Micron Millenia P5 - 133 MHz with 32 MB RAM, Windows 95 OS.