1. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Chapter 10 Summation Notation Section 10.1Section 10.1 Using Subscripts and Sigma Notation

2. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. 10.1 Using Subscripts and Sigma Notation Section 10.1

3. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Subscript Notation A survey of 9 local gas stations finds the following list of gas prices in dollars per gallon: 1.82, 1.84, 1.78, 1.85, 1.86, 1.87, 1.90, 1.88, 1.86. In working with a list like this it is often useful to use subscript notation, where we let g 1 stand for the first price in the list, g 2 for the second price, and so on up until g 9 for the last price. Thus, for example, we say g 4 = 1.85 to indicate that the fourth price is $1.85, or g 5 = g 9 to indicate that the prices at the fifth and ninth gas stations surveyed are the same ($1.86). A survey of 9 local gas stations finds the following list of gas prices in dollars per gallon: 1.82, 1.84, 1.78, 1.85, 1.86, 1.87, 1.90, 1.88, 1.86. In working with a list like this it is often useful to use subscript notation, where we let g 1 stand for the first price in the list, g 2 for the second price, and so on up until g 9 for the last price. Thus, for example, we say g 4 = 1.85 to indicate that the fourth price is $1.85, or g 5 = g 9 to indicate that the prices at the fifth and ninth gas stations surveyed are the same ($1.86). Section 10.1

4. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Example 1 Write expressions using subscript notation for (a) The cost of 5 gallons of gas at the second gas station. (b) The average price of gas at all stations. Solution (a) The cost of gas at the second station is g 2 dollars per gallon, so Cost of 5 gallons at second station = 5g 2. (b) The average price is obtained by adding up all the prices and dividing by the number of stations: We often shorten an expression like this using three dots to indicate the missing terms: Section 10.1

5. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Section 10.1 In the previous example there were 9 items in the list. Sometimes we use a letter to stand for the number of items. For example, we might ask a friend in another city to survey prices at their local gas stations. Assuming there are n stations surveyed, and using dots again to indicate missing values, we can list the prices as g 1, g 2,..., g n, where g i = Price of gas at the ith gas station, for i = 1, 2,..., n. Notice that in this case we have used a subscripted letter i to vary over the numbers from 1 to n. In the previous example there were 9 items in the list. Sometimes we use a letter to stand for the number of items. For example, we might ask a friend in another city to survey prices at their local gas stations. Assuming there are n stations surveyed, and using dots again to indicate missing values, we can list the prices as g 1, g 2,..., g n, where g i = Price of gas at the ith gas station, for i = 1, 2,..., n. Notice that in this case we have used a subscripted letter i to vary over the numbers from 1 to n.

6. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Example 2a There are n snowstorms in a town one winter. The first storm delivers d 1 inches of snow, the second delivers d 2 inches, and so on until the last storm, which delivers d n inches. (a) Write an expression for the total amount of snow that fell all winter. Solution (a) We have: Section 10.1

7. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Example 2b There are n snowstorms in a town one winter. The first storm delivers d 1 inches of snow, the second delivers d 2 inches, and so on until the last storm, which delivers d n inches. (b) Write an expression for the total amount of snow that fell in the final three storms. Solution (b) We have: Section 10.1 The word index is often used interchangeably with the term subscript.

8. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Sigma Notation To represent sums, we often use a special symbol, the capital Greek letter Σ (pronounced “sigma”). Using this notation, we write: The Σ tells us we are adding up some numbers. The a i tells us that the numbers we are adding are called a 1, a 2, and so on. The sum begins with a 1 and ends with a n because the subscript i starts at i = 1 (at the bottom of the Σ sign) and ends at i = n (at the top of the Σ sign). Sometimes the a i are given by a list, as in Example 1, and sometimes they are given by a formula in i, as in the next example. To represent sums, we often use a special symbol, the capital Greek letter Σ (pronounced “sigma”). Using this notation, we write: The Σ tells us we are adding up some numbers. The a i tells us that the numbers we are adding are called a 1, a 2, and so on. The sum begins with a 1 and ends with a n because the subscript i starts at i = 1 (at the bottom of the Σ sign) and ends at i = n (at the top of the Σ sign). Sometimes the a i are given by a list, as in Example 1, and sometimes they are given by a formula in i, as in the next example. Section 10.1

9. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Solution We have We find the terms in the sum by sequentially giving i the values 1 through 5 in the expression i 2. Section 10.1 Example 3 Write out the sum

10. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Section 10.1 Using summation notation we can write the solution to Example 1(b) as

11. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Example 4 There are n airlines. Airline i owns q i planes and carries p i passengers per plane, for i = 1, 2,..., n. Using sigma notation, write an expression for the average number of passengers per plane at all n airlines. Solution To find the average, we divide the total number of passengers by the total number of planes. Since airline i has q i planes, Since airline i carries p i passengers on each of its q i planes, it carries p i q i passengers in total. Thus So Section 10.1

12. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Example 5a Evaluate the following expressions based on Table 10.1, which gives the first 9 prime numbers. Table 10.1 Solution (a) Here, we find the terms by assigning successive integers starting at i = 1 and going up to i = 5. Then we add the terms: = 2 + 3 + 5 + 7 + 11 = 28. Section 10.1 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 p8p8 p9p9 23571113171923

13. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Example 5b Evaluate the following expressions based on Table 10.1, which gives the first 9 prime numbers. Table 10.1 Solution (b) Here, we are adding p i terms starting at i = 4 and going up to i = 8: = 7 + 11 + 13 + 17 + 19 = 67. Section 10.1 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 p8p8 p9p9 23571113171923

14. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Example 5c Evaluate the following expressions based on Table 10.1, which gives the first 9 prime numbers. Table 10.1 Solution (c) Here, we are adding 3 to the sum of p i terms starting at i = 1 and going up to i = 5: = 3 + 28 = 31. Section 10.1 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 p8p8 p9p9 23571113171923

15. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Example 5d Evaluate the following expressions based on Table 10.1, which gives the first 9 prime numbers. Table 10.1 Solution (d) Here, we are adding terms that look like 3 + p i, starting at i = 1 and going up to i = 5: = (3 + 2) + (3 + 3) + (3 + 5) + (3 + 7) + (3 + 11) = 5 + 6 + 8 + 10 + 14 = 43. (Solution 5d continued on next slide.) Section 10.1 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 p8p8 p9p9 23571113171923

16. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Example 5d (continued) Solution 5d continued Solution (d) Another approach would be to write regroup = 15 + 28 = 43. Notice that this is different from the answer we got in (c), so Section 10.1

17. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Example 5e Evaluate the following expressions based on Table 10.1, which gives the first 9 prime numbers. Table 10.1 Solution (e) Here, we are adding terms that look like 2p i, starting at i = 2 and going up to i = 4: = 2 · 3 + 2 · 5 + 2 · 7 = 30. Section 10.1 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 p8p8 p9p9 23571113171923

18. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Example 5f Evaluate the following expressions based on Table 10.1, which gives the first 9 prime numbers. Table 10.1 Solution (f) We are adding terms which alternate between negative and positive values: = −2 + 3 − 5 + 7 = 3. Section 10.1 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 p8p8 p9p9 23571113171923

19. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Example 6a Using sigma notation, write expressions standing for: (a) The square of the sum of the first 5 prime numbers. Solution (a) We have: The square of the sum of the first 5 prime numbers = (The sum of the first 5 prime numbers) 2 Section 10.1

20. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Example 6b Using sigma notation, write expressions standing for: (b) The sum of the squares of the first 5 prime numbers. Do these expressions have the same value? Solution (b) We have: The sum of the squares of the first 5 prime numbers = 4 + 9 + 25 + 49 + 121 = 208. The values are not the same. In general, Section 10.1

21. ALGEBRA: FORM AND FUNCTION 2 nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. 10.1 USING SUBSCRIPTS AND SIGMA NOTATION Key Points Subscript Notation Sigma Notation Section 10.1