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Warm Up: Section 2.11B Write a recursive routine for: (1). 6, 8, 10, 12, . . .

Warm Up: Section 2.11B Write a recursive routine for: (1). 6, 8, 10, 12, . . . (2). 1, 5, 9, 13, . . . Write an explicit formula for: (3). 10, 7, 4, 1, . . . (5). -6, -4, -2, . . . . Warm Up: Section 2.11B Write a recursive routine for:

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Warm Up: Section 2.11B Write a recursive routine for: (1). 6, 8, 10, 12, . . .

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  1. Warm Up: Section 2.11B Write a recursive routine for: (1). 6, 8, 10, 12, . . . (2). 1, 5, 9, 13, . . . Write an explicit formula for: (3). 10, 7, 4, 1, . . . (5). -6, -4, -2, . . .

  2. Warm Up: Section 2.11B Write a recursive routine for: (1). 6, 8, 10, 12, . . . an = an-1 + 2 with a1 = 6 (2). 1, 5, 9, 13, . . . an = an-1 + 4 with a1 = 1 Write an explicit formula for: (3). 10, 7, 4, 1, . . . an = 10 + (n – 1)(-3) (5). -6, -4, -2, . . . an = -6 + (n – 1)(2)

  3. Arithmetic Series Standard: MM2A3 d Essential Question: Can I evaluate and describe an arithmetic series? Section 2.11B

  4. Vocabulary: Series: the expression that results when the terms of a sequence are added together Sigma notation: another name for summation notation, which uses the Greek letter, sigma, written ∑ Arithmetic series: the expression formed by adding the terms of an arithmetic sequence, denotes by Sn

  5. Investigation 1: A series is the expression that results when the terms of a sequence are added together. Using your calculator, find the value of each series: (1). 3 + 7 + 11 + 15 + 19 = _________ (2). 14 + 34 + 54 + 74 = _________ 55 176

  6. To indicate a particular sum, the notation Sn can be used. S indicates summation and n identifies which terms are to be added. Thus S2 tells us to add the first two terms of the sequence. Calculate each indicated sum. (3). 5 + 9 + 13 + 17 + 21 a. S2 = ______ b. S4 = ______ c. S = ______ (4). 12 + 9 + 6 + 3 + 0 + (– 3) + (– 6) + (– 9) a. S2 = ______ b. S4= ______ c. S = ______ 14 44 65 12 21 30

  7. A series is sometimes written using sigma notation. Sigma is a Greek letter and is used to indicate a sum. The sigma notation is read as the sum of all terms ai for i from 1 to n. The sigma notation can also be written using the explicit formula for a sequence.

  8. Example: tells us to add the first 3 terms of a sequence where ai = 2i + 5. Calculate the value of the three terms: a1 = 2 + 5 = 7a2 = 4 + 5 = 9a3 = 6 + 5 = 11 Now add the terms together: = 7 + 9 + 11 = 27.

  9. Find each sum: (5). = ____ + ____ + ____ + ____ = _____ (6). = ___ + ___ + ___ + ___ + ___ = ____ 30 3 6 9 12 -1 3 2 1 0 5

  10. Consider the sequence 5, 11, 17, 23, 29, … An explicit formula for any term an of this sequence is an = 5 + 6(n – 1). If we wanted to write a series for this sequence, we would use the following notation: To find an explicit formula for the series: 5 + 11 + 17 + 23 we write

  11. Use the explicit formula for the sequence to write a formula for each series using sigma notation. Recall: Yesterday, we learned the explicit formula for an arithmetic sequence: an = a1 + (n – 1)d 5 [5 + (i – 1)4] (7). 5 + 9 + 13 + 17 + 21 = (8). 12 + 9 + 6 + 3 + 0 + (– 3) + (– 6) + (– 9) = i=1 8 [12 + (i – 1)(-3)] i=1

  12. Check for Understanding: (9). Find S3 for the sequence 7, 10, 13, 15, . . . (10). Find the sum (11). Write the sigma notation for the series: -2 + 3 + 8 + 13 + 18 + 23 = S3 = 7 + 10 + 13 = 30 = 6 + 7 + 8 = 21 6 [-2 + (i – 1)5] i=1

  13. Investigation 2: For an arithmetic series with n terms, the sum of the first n terms is Note: Remember that an = a1 + (n – 1)d.

  14. What type of equation is contained in the box?______________ Any sequence of partial sums of an arithmetic sequence is an example of a quadratic function because n is always raised to the second power. This formula allows us to find a sum without identifying each term in the series!!! Quadratic

  15. Calculate each sum using the formula for the sum of an arithmetic series: (12). = a1 = 2 + 5 = 7 an= a8 = 16 + 5 = 21 n = 8

  16. (13). a1 = 1 – 3 = -2 an= a10 = 10 – 3 = 7 n = 10

  17. (14). a1 = 3/4 an= a15 = (3/4)(15) = 45/4 n = 15

  18. Write the summation notation for each series, then find the sum: (15). 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 a1 = 1 an= a10 = 19 n = 10

  19. Write the summation notation for each series, then find the sum: (16). 12 + 9 + 6 + 3 + 0 + (– 3) + (– 6) + (– 9) a1 = 12 an= a8 = -9 n = 8

  20. Write the summation notation for each series, then find the sum: (17). 5 + 9 + 13 +17 + 21 + ... + 49 + 53 How many terms are there in the summation? an= 5 + (n – 1)4 53 = 5 + 4n – 4 53 = 4n + 1 52 = 4n 13 = n

  21. Write the summation notation for each series, then find the sum: (17). 5 + 9 + 13 +17 + 21 + ... + 49 + 53 a1 = 5 an= a13 = 53 n = 13

  22. Write the summation notation for each series, then find the sum: (18). 1 + 5 + 9 +13 + 17 + ... (Find S50 ) a1 = 1 an= a50 = 197 n = 50

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