Arithmetic Series

1. Arithmetic Series Section 9.1

2. According to the U.S. Environmental Protection Agency, each American produced an average of 978 lb of trash in 1960. • This increased to 1336 lb in 1980. By 2000, trash production had risen to 1646 lb/yr per person. • You have learned in previous chapters how to write a sequence to describe the amount of trash produced per person each year. If you wanted to find the total amount of trash a person produced in his or her lifetime, you would add the numbers in this sequence.

4. Finding the value of a series is a problem that has intrigued mathematicians for centuries. Chinese mathematician Chu Shih-chieh called the sum 1+2+3+ . . . + n a “pile of reeds” because it can be pictured like the diagram at right. The diagram shows S9, the sum of the first nine terms of this sequence, 1+2+3+. . .+9. The sum of any finite, or limited, number of terms is called a partial sum of the series.

6. How could you find the sum of the integers 1 through 100? • The most obvious method is to add the terms, one by one. You can use technology and a recursive formula to do this quickly.

7. In column A, create the sequence 1, 2, 3, . . . , and name it n.

8. In column B, create the sequence that you are going to sum and name it sequence.

9. Name the third list partial_sum. • In cell C1, type the initial term of partial_sum. • In C2, type = c1 + b2, press enter, choose Data⎮ Fill Down, select the cells you want to fill, and press enter.

10. To graph the partial sums, create a scatter plot of (n, partialsum) in either Graphs & Geometry or Data & Statistics. You may need to adjust the window settings to view larger terms.

11. When you compute this sum recursively, you or the calculator must compute each of the individual terms.

12. The investigation will give you an opportunity to discover at least one explicit formula for calculating the partial sum of an arithmetic series without finding all terms and adding.

13. Arithmetic Series Formula • Step 1: The lengths of the rows of this step-shaped figure represent terms of an arithmetic sequence. Write the sequence u1, u2, u3, u4, u5, represented by the figure. What is the sum of the series?

14. Step 2: If you cut out two copies of this figure and slid them together to make a rectangle, what would the dimensions of your rectangle be? • Step 3: Have each member of your group create a different arithmetic sequence. Each of you can choose a starting value and a common difference. (Use positive numbers less than 10 for each of these.) On graph paper, draw two copies of a step-shaped figure representing your sequence.

15. Step 4: Cut out both copies of the step-shaped figure. Slide the two congruent shapes together to form a rectangle, and then calculate the dimensions and area of the rectangle. Now express this area in terms of the number of rows, n, and the first and last terms of the sequence. • Step 5: Based on what you have discovered, what is a formula for the partial sum, Sn, of an arithmetic series? Describe the relationship between your formula and the dimensions of your rectangle.

16. Example • Find the sum of the integers 1 through 100, without using a calculator. • Carl Friedrich Gauss solved this problem by adding the terms in pairs. Consider the series written in ascending and descending order, as shown. The sum of every column is 101, and there are 100 columns. Thus, the sum of the integers 1 through 100 is

17. You can extend the method in the example to any arithmetic series. Before continuing, take a moment to consider why the sum of the reeds in the original pile can be calculated using the expression • What do the 9, 1, 9, and 2 represent in this context?