Transcript ch10x
Chapter 10 Summation Notation Section 10.1 Using Subscripts and Sigma Notation ALGEBRA: FORM AND FUNCTION 2nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. 10.1 Using Subscripts and Sigma Notation Section 10.1 ALGEBRA: FORM AND FUNCTION 2nd 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 g1 stand for the first price in the list, g2 for the second price, and so on up until g9 for the last price. Thus, for example, we say g4 = 1.85 to indicate that the fourth price is $1.85, or g5 = g9 to indicate that the prices at the fifth and ninth gas stations surveyed are the same ($1.86). Section 10.1 ALGEBRA: FORM AND FUNCTION 2nd 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 g2 dollars per gallon, so Cost of 5 gallons at second station = 5g2. (b) The average price is obtained by adding up all the prices and dividing by the number of stations: g1 g2 g3 g4 g5 g6 g7 g8 g9 Average cost . 9 We often shorten an expression like this using three dots to indicate the missing terms: g1 g2 Average cost 9 Section 10.1 g9 . ALGEBRA: FORM AND FUNCTION 2nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. 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 gi = 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. Section 10.1 ALGEBRA: FORM AND FUNCTION 2nd 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 d1 inches of snow, the second delivers d2 inches, and so on until the last storm, which delivers dn inches. (a) Write an expression for the total amount of snow that fell all winter. Solution (a) We have: Total amount of snow Amount in Amount in first storm second storm d1 d1 d2 Section 10.1 d2 Amount in last storm dn dn . ALGEBRA: FORM AND FUNCTION 2nd 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 d1 inches of snow, the second delivers d2 inches, and so on until the last storm, which delivers dn inches. (b) Write an expression for the total amount of snow that fell in the final three storms. Solution (b) We have: Total amount of snow Amount in Amount in in final three storms first storm second storm dn dn dn1 dn2 . dn1 Amount in last storm dn2 The word index is often used interchangeably with the term subscript. Section 10.1 ALGEBRA: FORM AND FUNCTION 2nd 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: n a i 1 i a1 a2 a3 an . The Σ tells us we are adding up some numbers. The ai tells us that the numbers we are adding are called a1, a2, and so on. The sum begins with a1 and ends with an 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 ai 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 ALGEBRA: FORM AND FUNCTION 2nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Example 3 5 Write out the sum i . 2 i 1 Solution We have 5 2 2 2 2 2 2 i 1 2 3 4 5 . i 1 We find the terms in the sum by sequentially giving i the values 1 through 5 in the expression i2. Section 10.1 ALGEBRA: FORM AND FUNCTION 2nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Using summation notation we can write the solution to Example 1(b) as g1 g2 g9 1 1 9 Average price ( g1 g2 g9 ) gi . 9 9 9 i 1 Section 10.1 ALGEBRA: FORM AND FUNCTION 2nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc. Example 4 There are n airlines. Airline i owns qi planes and carries pi 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 qi planes, n Total number of planes from all airlines qi . i 1 Since airline i carries pi passengers on each of its qi planes, it carries piqi passengers in total. Thus n Total number of passengers from all airlines pi qi . i 1 So Total number of passengers per plane Section 10.1 n i 1 n pi qi q . i 1 i ALGEBRA: FORM AND FUNCTION 2nd 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 5 gives the first 9 prime numbers. (a) pi i 1 Table 10.1 p1 p2 p3 p4 p5 p6 p7 p8 p9 2 3 5 7 11 13 17 19 23 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: 5 p i 1 i p1 p2 p3 p4 p5 = 2 + 3 + 5 + 7 + 11 = 28. Section 10.1 ALGEBRA: FORM AND FUNCTION 2nd 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 8 gives the first 9 prime numbers. (b) pi i 4 Table 10.1 p1 p2 p3 p4 p5 p6 p7 p8 p9 2 3 5 7 11 13 17 19 23 Solution (b) Here, we are adding pi terms starting at i = 4 and going up to i = 8: 8 p i 4 i p4 p5 p6 p7 p8 = 7 + 11 + 13 + 17 + 19 = 67. Section 10.1 ALGEBRA: FORM AND FUNCTION 2nd 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 5 gives the first 9 prime numbers. (c) 3 pi i 1 Table 10.1 p1 p2 p3 p4 p5 p6 p7 p8 p9 2 3 5 7 11 13 17 19 23 Solution (c) Here, we are adding 3 to the sum of pi terms starting at i = 1 and going up to i = 5: 5 from part (a) equals 28 3 pi 3 p1 p2 p3 p4 p5 i 1 = 3 + 28 = 31. Section 10.1 ALGEBRA: FORM AND FUNCTION 2nd 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 5 gives the first 9 prime numbers. (d) (3 pi ) i 1 Table 10.1 p1 p2 p3 p4 p5 p6 p7 p8 p9 2 3 5 7 11 13 17 19 23 Solution (d) Here, we are adding terms that look like 3 + pi, starting at i = 1 and going up to i = 5: 5 (3 p ) (3 p ) (3 p ) i 1 i 1 2 (3 p5 ) = (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 ALGEBRA: FORM AND FUNCTION 2nd 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 5 (3 p ) (3 p ) (3 p ) i 1 i 1 2 (3 p5 ) from part (a) equals 28 (3 3 3 3 3) p1 p2 p3 p4 p5 regroup = 15 + 28 = 43. Notice that this is different from the answer we got in (c), so 5 3 pi i 1 Section 10.1 5 is not equivalent to (3 p ) i 1 i ALGEBRA: FORM AND FUNCTION 2nd 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 4 gives the first 9 prime numbers. (e) 2 pi i 2 Table 10.1 p1 p2 p3 p4 p5 p6 p7 p8 p9 2 3 5 7 11 13 17 19 23 Solution (e) Here, we are adding terms that look like 2pi, starting at i = 2 and going up to i = 4: 4 2p i 2 i 2 p2 2 p3 2 p4 = 2 · 3 + 2 · 5 + 2 · 7 = 30. Section 10.1 ALGEBRA: FORM AND FUNCTION 2nd 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 4 gives the first 9 prime numbers. i (f) ( 1) p i 1 Table 10.1 i p1 p2 p3 p4 p5 p6 p7 p8 p9 2 3 5 7 11 13 17 19 23 Solution (f) We are adding terms which alternate between negative and positive values: 4 i ( 1) pi p1 p2 p3 p4 i 1 = −2 + 3 − 5 + 7 = 3. Section 10.1 ALGEBRA: FORM AND FUNCTION 2nd 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 2 2 2 p (2 3 5 7 11) 28 784. i k 1 5 Section 10.1 ALGEBRA: FORM AND FUNCTION 2nd 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 5 p1 p2 p3 p4 p5 pi 2 2 2 32 52 7 2 112 2 2 2 2 2 k 1 = 4 + 9 + 25 + 49 + 121 = 208. The values are not the same. In general, 2 n n 2 p p . i i k 1 k 1 Section 10.1 ALGEBRA: FORM AND FUNCTION 2nd 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 ALGEBRA: FORM AND FUNCTION 2nd edition by McCallum, Connally, Hughes-Hallett, et al.,Copyright 2015, John Wiley & Sons, Inc.